[{"problem_text": "Find the effective annual yield of a bank account that pays interest at a rate of 7%, compounded daily; that is, divide the difference between the final and initial balances by the initial balance.", "answer_latex": "7.25", "answer_number": "7.25", "unit": " %", "source": "diff", "problemid": " page 130-7", "comment": " ", "solution": ""}, {"problem_text": "Consider a tank used in certain hydrodynamic experiments. After one experiment the tank contains $200 \\mathrm{~L}$ of a dye solution with a concentration of $1 \\mathrm{~g} / \\mathrm{L}$. To prepare for the next experiment, the tank is to be rinsed with fresh water flowing in at a rate of $2 \\mathrm{~L} / \\mathrm{min}$, the well-stirred solution flowing out at the same rate. Find the time that will elapse before the concentration of dye in the tank reaches $1 \\%$ of its original value.", "answer_latex": " 460.5", "answer_number": "460.5", "unit": " min", "source": "diff", "problemid": " Page 59-1", "comment": " ", "solution": ""}, {"problem_text": "A certain vibrating system satisfies the equation $u^{\\prime \\prime}+\\gamma u^{\\prime}+u=0$. Find the value of the damping coefficient $\\gamma$ for which the quasi period of the damped motion is $50 \\%$ greater than the period of the corresponding undamped motion.", "answer_latex": " 1.4907", "answer_number": "1.4907", "unit": " ", "source": "diff", "problemid": " page203-13", "comment": " ", "solution": ""}, {"problem_text": "Find the value of $y_0$ for which the solution of the initial value problem\r\n$$\r\ny^{\\prime}-y=1+3 \\sin t, \\quad y(0)=y_0\r\n$$\r\nremains finite as $t \\rightarrow \\infty$", "answer_latex": " -2.5", "answer_number": " -2.5", "unit": " ", "source": "diff", "problemid": " Page 40-30", "comment": " ", "solution": ""}, {"problem_text": "A certain spring-mass system satisfies the initial value problem\r\n$$\r\nu^{\\prime \\prime}+\\frac{1}{4} u^{\\prime}+u=k g(t), \\quad u(0)=0, \\quad u^{\\prime}(0)=0\r\n$$\r\nwhere $g(t)=u_{3 / 2}(t)-u_{5 / 2}(t)$ and $k>0$ is a parameter.\r\nSuppose $k=2$. Find the time $\\tau$ after which $|u(t)|<0.1$ for all $t>\\tau$.", "answer_latex": " 25.6773", "answer_number": "25.6773", "unit": " ", "source": "diff", "problemid": " page336-16", "comment": " ", "solution": ""}, {"problem_text": "Suppose that a sum $S_0$ is invested at an annual rate of return $r$ compounded continuously.\r\nDetermine $T$ if $r=7 \\%$.", "answer_latex": " 9.90", "answer_number": "9.90", "unit": " year", "source": "diff", "problemid": " page 60-7", "comment": " ", "solution": ""}, {"problem_text": "A mass weighing $2 \\mathrm{lb}$ stretches a spring 6 in. If the mass is pulled down an additional 3 in. and then released, and if there is no damping, by determining the position $u$ of the mass at any time $t$, find the frequency of the motion", "answer_latex": " $\\pi/4$", "answer_number": "0.7854", "unit": " s", "source": "diff", "problemid": " page202-5", "comment": " ", "solution": ""}, {"problem_text": "If $\\mathbf{x}=\\left(\\begin{array}{c}2 \\\\ 3 i \\\\ 1-i\\end{array}\\right)$ and $\\mathbf{y}=\\left(\\begin{array}{c}-1+i \\\\ 2 \\\\ 3-i\\end{array}\\right)$, find $(\\mathbf{y}, \\mathbf{y})$.", "answer_latex": " 16", "answer_number": "16", "unit": " ", "source": "diff", "problemid": " page372-8", "comment": " ", "solution": ""}, {"problem_text": "Consider the initial value problem\r\n$$\r\n4 y^{\\prime \\prime}+12 y^{\\prime}+9 y=0, \\quad y(0)=1, \\quad y^{\\prime}(0)=-4 .\r\n$$\r\nDetermine where the solution has the value zero.", "answer_latex": " 0.4", "answer_number": " 0.4", "unit": " ", "source": "diff", "problemid": " page172-15", "comment": " ", "solution": ""}, {"problem_text": "A certain college graduate borrows $8000 to buy a car. The lender charges interest at an annual rate of 10%. What monthly payment rate is required to pay off the loan in 3 years?", "answer_latex": " 258.14", "answer_number": " 258.14", "unit": " $", "source": "diff", "problemid": " page131-9", "comment": " ", "solution": ""}, {"problem_text": "Consider the initial value problem\r\n$$\r\ny^{\\prime \\prime}+\\gamma y^{\\prime}+y=k \\delta(t-1), \\quad y(0)=0, \\quad y^{\\prime}(0)=0\r\n$$\r\nwhere $k$ is the magnitude of an impulse at $t=1$ and $\\gamma$ is the damping coefficient (or resistance).\r\nLet $\\gamma=\\frac{1}{2}$. Find the value of $k$ for which the response has a peak value of 2 ; call this value $k_1$.", "answer_latex": " 2.8108", "answer_number": "2.8108", "unit": " ", "source": "diff", "problemid": " page344-15", "comment": " ", "solution": ""}, {"problem_text": "If a series circuit has a capacitor of $C=0.8 \\times 10^{-6} \\mathrm{~F}$ and an inductor of $L=0.2 \\mathrm{H}$, find the resistance $R$ so that the circuit is critically damped.", "answer_latex": " 1000", "answer_number": "1000", "unit": " $\\Omega$", "source": "diff", "problemid": " page203-18", "comment": " ", "solution": ""}, {"problem_text": "If $y_1$ and $y_2$ are a fundamental set of solutions of $t y^{\\prime \\prime}+2 y^{\\prime}+t e^t y=0$ and if $W\\left(y_1, y_2\\right)(1)=2$, find the value of $W\\left(y_1, y_2\\right)(5)$.", "answer_latex": " 2/25", "answer_number": "0.08", "unit": " ", "source": "diff", "problemid": " page156-34", "comment": " ", "solution": ""}, {"problem_text": "Consider the initial value problem\r\n$$\r\n5 u^{\\prime \\prime}+2 u^{\\prime}+7 u=0, \\quad u(0)=2, \\quad u^{\\prime}(0)=1\r\n$$\r\nFind the smallest $T$ such that $|u(t)| \\leq 0.1$ for all $t>T$.", "answer_latex": "14.5115", "answer_number": "14.5115", "unit": " ", "source": "diff", "problemid": " page163-24", "comment": " ", "solution": ""}, {"problem_text": "Consider the initial value problem\r\n$$\r\ny^{\\prime}=t y(4-y) / 3, \\quad y(0)=y_0\r\n$$\r\nSuppose that $y_0=0.5$. Find the time $T$ at which the solution first reaches the value 3.98.", "answer_latex": " 3.29527", "answer_number": "3.29527", "unit": " ", "source": "diff", "problemid": "Page 49 27 ", "comment": " ", "solution": ""}, {"problem_text": "Consider the initial value problem\r\n$$\r\n2 y^{\\prime \\prime}+3 y^{\\prime}-2 y=0, \\quad y(0)=1, \\quad y^{\\prime}(0)=-\\beta,\r\n$$\r\nwhere $\\beta>0$.\r\nFind the smallest value of $\\beta$ for which the solution has no minimum point.", "answer_latex": " 2", "answer_number": "2", "unit": " ", "source": "diff", "problemid": " page144-25", "comment": " ", "solution": ""}, {"problem_text": "Consider the initial value problem\r\n$$\r\ny^{\\prime}=t y(4-y) /(1+t), \\quad y(0)=y_0>0 .\r\n$$\r\nIf $y_0=2$, find the time $T$ at which the solution first reaches the value 3.99.", "answer_latex": " 2.84367", "answer_number": "2.84367", "unit": " ", "source": "diff", "problemid": " Page 49 28", "comment": " ", "solution": ""}, {"problem_text": "A mass of $0.25 \\mathrm{~kg}$ is dropped from rest in a medium offering a resistance of $0.2|v|$, where $v$ is measured in $\\mathrm{m} / \\mathrm{s}$.\r\nIf the mass is to attain a velocity of no more than $10 \\mathrm{~m} / \\mathrm{s}$, find the maximum height from which it can be dropped.", "answer_latex": " 13.45", "answer_number": " 13.45", "unit": " m", "source": "diff", "problemid": "page66-28", "comment": " ", "solution": ""}, {"problem_text": "A home buyer can afford to spend no more than $\\$ 800$ /month on mortgage payments. Suppose that the interest rate is $9 \\%$ and that the term of the mortgage is 20 years. Assume that interest is compounded continuously and that payments are also made continuously.\r\nDetermine the maximum amount that this buyer can afford to borrow.", "answer_latex": " 89,034.79", "answer_number": "89,034.79", "unit": " $", "source": "diff", "problemid": " page 61-10", "comment": " ", "solution": ""}, {"problem_text": "A spring is stretched 6 in by a mass that weighs $8 \\mathrm{lb}$. The mass is attached to a dashpot mechanism that has a damping constant of $0.25 \\mathrm{lb} \\cdot \\mathrm{s} / \\mathrm{ft}$ and is acted on by an external force of $4 \\cos 2 t \\mathrm{lb}$.\r\nIf the given mass is replaced by a mass $m$, determine the value of $m$ for which the amplitude of the steady state response is maximum.", "answer_latex": " 4", "answer_number": "4", "unit": " slugs", "source": "diff", "problemid": " page216-11", "comment": " ", "solution": ""}, {"problem_text": "A recent college graduate borrows $\\$ 100,000$ at an interest rate of $9 \\%$ to purchase a condominium. Anticipating steady salary increases, the buyer expects to make payments at a monthly rate of $800(1+t / 120)$, where $t$ is the number of months since the loan was made.\r\nAssuming that this payment schedule can be maintained, when will the loan be fully paid?", "answer_latex": " 135.36", "answer_number": " 135.36", "unit": " months", "source": "diff", "problemid": " page61-11", "comment": " ", "solution": ""}, {"problem_text": "Consider the initial value problem\r\n$$\r\ny^{\\prime}+\\frac{1}{4} y=3+2 \\cos 2 t, \\quad y(0)=0\r\n$$\r\nDetermine the value of $t$ for which the solution first intersects the line $y=12$.", "answer_latex": " 10.065778", "answer_number": "10.065778", "unit": " ", "source": "diff", "problemid": " Page 40 29", "comment": " ", "solution": ""}, {"problem_text": "An investor deposits $1000 in an account paying interest at a rate of 8% compounded monthly, and also makes additional deposits of \\$25 per month. Find the balance in the account after 3 years.", "answer_latex": " 2283.63", "answer_number": "2283.63", "unit": " $", "source": "diff", "problemid": " page 131-8", "comment": " ", "solution": ""}, {"problem_text": "A mass of $0.25 \\mathrm{~kg}$ is dropped from rest in a medium offering a resistance of $0.2|v|$, where $v$ is measured in $\\mathrm{m} / \\mathrm{s}$.\r\nIf the mass is dropped from a height of $30 \\mathrm{~m}$, find its velocity when it hits the ground.", "answer_latex": " 11.58", "answer_number": " 11.58", "unit": " m/s", "source": "diff", "problemid": " page 66-28", "comment": " ", "solution": ""}, {"problem_text": "A mass of $100 \\mathrm{~g}$ stretches a spring $5 \\mathrm{~cm}$. If the mass is set in motion from its equilibrium position with a downward velocity of $10 \\mathrm{~cm} / \\mathrm{s}$, and if there is no damping, determine when does the mass first return to its equilibrium position.", "answer_latex": " $\\pi/14$", "answer_number": "0.2244", "unit": " s", "source": "diff", "problemid": " page202-6", "comment": " ", "solution": ""}, {"problem_text": "Suppose that a tank containing a certain liquid has an outlet near the bottom. Let $h(t)$ be the height of the liquid surface above the outlet at time $t$. Torricelli's principle states that the outflow velocity $v$ at the outlet is equal to the velocity of a particle falling freely (with no drag) from the height $h$.\r\nConsider a water tank in the form of a right circular cylinder that is $3 \\mathrm{~m}$ high above the outlet. The radius of the tank is $1 \\mathrm{~m}$ and the radius of the circular outlet is $0.1 \\mathrm{~m}$. If the tank is initially full of water, determine how long it takes to drain the tank down to the level of the outlet.", "answer_latex": " 130.41", "answer_number": "130.41", "unit": " s", "source": "diff", "problemid": "page 60-6", "comment": " ", "solution": ""}, {"problem_text": "Solve the initial value problem $y^{\\prime \\prime}-y^{\\prime}-2 y=0, y(0)=\\alpha, y^{\\prime}(0)=2$. Then find $\\alpha$ so that the solution approaches zero as $t \\rightarrow \\infty$.", "answer_latex": " \u22122", "answer_number": "\u22122", "unit": " ", "source": "diff", "problemid": " page144-21", "comment": " ", "solution": ""}, {"problem_text": "If $y_1$ and $y_2$ are a fundamental set of solutions of $t^2 y^{\\prime \\prime}-2 y^{\\prime}+(3+t) y=0$ and if $W\\left(y_1, y_2\\right)(2)=3$, find the value of $W\\left(y_1, y_2\\right)(4)$.", "answer_latex": " 4.946", "answer_number": "4.946", "unit": " ", "source": "diff", "problemid": " page156-35", "comment": " ", "solution": ""}, {"problem_text": " Radium-226 has a half-life of 1620 years. Find the time period during which a given amount of this material is reduced by one-quarter.", "answer_latex": " 672.4", "answer_number": " 672.4", "unit": " Year", "source": "diff", "problemid": " Page 17 14", "comment": " ", "solution": ""}, {"problem_text": "A tank originally contains $100 \\mathrm{gal}$ of fresh water. Then water containing $\\frac{1}{2} \\mathrm{lb}$ of salt per gallon is poured into the tank at a rate of $2 \\mathrm{gal} / \\mathrm{min}$, and the mixture is allowed to leave at the same rate. After $10 \\mathrm{~min}$ the process is stopped, and fresh water is poured into the tank at a rate of $2 \\mathrm{gal} / \\mathrm{min}$, with the mixture again leaving at the same rate. Find the amount of salt in the tank at the end of an additional $10 \\mathrm{~min}$.", "answer_latex": " 7.42", "answer_number": " 7.42", "unit": " lb", "source": "diff", "problemid": "Page 60-3 ", "comment": " ", "solution": ""}, {"problem_text": "A young person with no initial capital invests $k$ dollars per year at an annual rate of return $r$. Assume that investments are made continuously and that the return is compounded continuously.\r\nIf $r=7.5 \\%$, determine $k$ so that $\\$ 1$ million will be available for retirement in 40 years.", "answer_latex": " 3930", "answer_number": "3930", "unit": " $", "source": "diff", "problemid": " page 60-8", "comment": " ", "solution": ""}, {"problem_text": "Consider the initial value problem\r\n$$\r\ny^{\\prime \\prime}+2 a y^{\\prime}+\\left(a^2+1\\right) y=0, \\quad y(0)=1, \\quad y^{\\prime}(0)=0 .\r\n$$\r\nFor $a=1$ find the smallest $T$ such that $|y(t)|<0.1$ for $t>T$.", "answer_latex": "1.8763", "answer_number": "1.8763", "unit": " ", "source": "diff", "problemid": " page164-26", "comment": " ", "solution": ""}, {"problem_text": "Consider the initial value problem\r\n$$\r\ny^{\\prime \\prime}+\\gamma y^{\\prime}+y=\\delta(t-1), \\quad y(0)=0, \\quad y^{\\prime}(0)=0,\r\n$$\r\nwhere $\\gamma$ is the damping coefficient (or resistance).\r\nFind the time $t_1$ at which the solution attains its maximum value.", "answer_latex": " 2.3613", "answer_number": "2.3613", "unit": " ", "source": "diff", "problemid": " page344-14", "comment": " ", "solution": ""}, {"problem_text": "Consider the initial value problem\r\n$$\r\ny^{\\prime}+\\frac{2}{3} y=1-\\frac{1}{2} t, \\quad y(0)=y_0 .\r\n$$\r\nFind the value of $y_0$ for which the solution touches, but does not cross, the $t$-axis.", "answer_latex": " \u22121.642876", "answer_number": "\u22121.642876", "unit": " ", "source": "diff", "problemid": " Page 40 28", "comment": " ", "solution": ""}, {"problem_text": "A radioactive material, such as the isotope thorium-234, disintegrates at a rate proportional to the amount currently present. If $Q(t)$ is the amount present at time $t$, then $d Q / d t=-r Q$, where $r>0$ is the decay rate. If $100 \\mathrm{mg}$ of thorium-234 decays to $82.04 \\mathrm{mg}$ in 1 week, determine the decay rate $r$.", "answer_latex": " 0.02828", "answer_number": "0.02828", "unit": " $\\text{day}^{-1}$", "source": "diff", "problemid": " Section 1.2, page 15 12. (a)", "comment": " ", "solution": ""}, {"problem_text": "Newton's law of cooling states that the temperature of an object changes at a rate proportional to the difference between its temperature and that of its surroundings. Suppose that the temperature of a cup of coffee obeys Newton's law of cooling. If the coffee has a temperature of $200^{\\circ} \\mathrm{F}$ when freshly poured, and $1 \\mathrm{~min}$ later has cooled to $190^{\\circ} \\mathrm{F}$ in a room at $70^{\\circ} \\mathrm{F}$, determine when the coffee reaches a temperature of $150^{\\circ} \\mathrm{F}$.", "answer_latex": " 6.07", "answer_number": " 6.07", "unit": " min", "source": "diff", "problemid": " page62-16", "comment": " ", "solution": ""}, {"problem_text": "Solve the initial value problem $4 y^{\\prime \\prime}-y=0, y(0)=2, y^{\\prime}(0)=\\beta$. Then find $\\beta$ so that the solution approaches zero as $t \\rightarrow \\infty$.", "answer_latex": " -1", "answer_number": " -1", "unit": " ", "source": "diff", "problemid": " page144-22", "comment": " ", "solution": ""}, {"problem_text": "Consider the initial value problem (see Example 5)\r\n$$\r\ny^{\\prime \\prime}+5 y^{\\prime}+6 y=0, \\quad y(0)=2, \\quad y^{\\prime}(0)=\\beta\r\n$$\r\nwhere $\\beta>0$.\r\nDetermine the smallest value of $\\beta$ for which $y_m \\geq 4$.", "answer_latex": " 16.3923", "answer_number": "16.3923", "unit": " ", "source": "diff", "problemid": " page145-26", "comment": " ", "solution": ""}, {"problem_text": "A home buyer can afford to spend no more than $\\$ 800 /$ month on mortgage payments. Suppose that the interest rate is $9 \\%$ and that the term of the mortgage is 20 years. Assume that interest is compounded continuously and that payments are also made continuously.\r\nDetermine the total interest paid during the term of the mortgage.", "answer_latex": " 102,965.21", "answer_number": "102,965.21", "unit": " $", "source": "diff", "problemid": " page61-10", "comment": " ", "solution": ""}, {"problem_text": "Find the fundamental period of the given function:\r\n$$f(x)=\\left\\{\\begin{array}{ll}(-1)^n, & 2 n-1 \\leq x<2 n, \\\\ 1, & 2 n \\leq x<2 n+1 ;\\end{array} \\quad n=0, \\pm 1, \\pm 2, \\ldots\\right.$$", "answer_latex": " 4", "answer_number": "4", "unit": " ", "source": "diff", "problemid": " page593-8", "comment": " ", "solution": ""}, {"problem_text": "A homebuyer wishes to finance the purchase with a \\$95,000 mortgage with a 20-year term. What is the maximum interest rate the buyer can afford if the monthly payment is not to exceed \\$900?", "answer_latex": " 9.73", "answer_number": " 9.73", "unit": " %", "source": "diff", "problemid": " page131-13", "comment": " ", "solution": ""}, {"problem_text": "A homebuyer wishes to take out a mortgage of $100,000 for a 30-year period. What monthly payment is required if the interest rate is 9%?", "answer_latex": "804.62", "answer_number": "804.62", "unit": "$", "source": "diff", "problemid": " page131-10", "comment": " ", "solution": ""}, {"problem_text": "Let a metallic rod $20 \\mathrm{~cm}$ long be heated to a uniform temperature of $100^{\\circ} \\mathrm{C}$. Suppose that at $t=0$ the ends of the bar are plunged into an ice bath at $0^{\\circ} \\mathrm{C}$, and thereafter maintained at this temperature, but that no heat is allowed to escape through the lateral surface. Determine the temperature at the center of the bar at time $t=30 \\mathrm{~s}$ if the bar is made of silver.", "answer_latex": " 35.91", "answer_number": " 35.91", "unit": " ${ }^{\\circ} \\mathrm{C}$", "source": "diff", "problemid": " page619-18", "comment": " ", "solution": ""}, {"problem_text": "Find $\\gamma$ so that the solution of the initial value problem $x^2 y^{\\prime \\prime}-2 y=0, y(1)=1, y^{\\prime}(1)=\\gamma$ is bounded as $x \\rightarrow 0$.", "answer_latex": " 2", "answer_number": "2", "unit": " ", "source": "diff", "problemid": " page277-37", "comment": " ", "solution": ""}, {"problem_text": "A tank contains 100 gal of water and $50 \\mathrm{oz}$ of salt. Water containing a salt concentration of $\\frac{1}{4}\\left(1+\\frac{1}{2} \\sin t\\right) \\mathrm{oz} / \\mathrm{gal}$ flows into the tank at a rate of $2 \\mathrm{gal} / \\mathrm{min}$, and the mixture in the tank flows out at the same rate.\r\nThe long-time behavior of the solution is an oscillation about a certain constant level. What is the amplitude of the oscillation?", "answer_latex": " 0.24995", "answer_number": "0.24995", "unit": " ", "source": "diff", "problemid": " Page 60-5", "comment": " ", "solution": ""}, {"problem_text": "A mass weighing $8 \\mathrm{lb}$ stretches a spring 1.5 in. The mass is also attached to a damper with coefficient $\\gamma$. Determine the value of $\\gamma$ for which the system is critically damped; be sure to give the units for $\\gamma$", "answer_latex": "8", "answer_number": "8", "unit": " $\\mathrm{lb} \\cdot \\mathrm{s} / \\mathrm{ft}$", "source": "diff", "problemid": " page203-17", "comment": " ", "solution": ""}, {"problem_text": "Your swimming pool containing 60,000 gal of water has been contaminated by $5 \\mathrm{~kg}$ of a nontoxic dye that leaves a swimmer's skin an unattractive green. The pool's filtering system can take water from the pool, remove the dye, and return the water to the pool at a flow rate of $200 \\mathrm{gal} / \\mathrm{min}$. Find the time $T$ at which the concentration of dye first reaches the value $0.02 \\mathrm{~g} / \\mathrm{gal}$.", "answer_latex": " 7.136", "answer_number": "7.136", "unit": " hour", "source": "diff", "problemid": " Page 18 19", "comment": " ", "solution": ""}, {"problem_text": "For small, slowly falling objects, the assumption made in the text that the drag force is proportional to the velocity is a good one. For larger, more rapidly falling objects, it is more accurate to assume that the drag force is proportional to the square of the velocity.  If m = 10 kg, find the drag coefficient so that the limiting velocity is 49 m/s.", "answer_latex": "$\\frac{2}{49}$", "answer_number": "0.0408", "unit": " ", "source": "diff", "problemid": " 1 25(c)", "comment": " ", "solution": ""}, {"problem_text": "Consider the initial value problem\r\n$$\r\n3 u^{\\prime \\prime}-u^{\\prime}+2 u=0, \\quad u(0)=2, \\quad u^{\\prime}(0)=0\r\n$$\r\nFor $t>0$ find the first time at which $|u(t)|=10$.", "answer_latex": " 10.7598", "answer_number": " 10.7598", "unit": " ", "source": "diff", "problemid": " page163-23", "comment": " ", "solution": ""}, {"problem_text": "Consider the initial value problem\r\n$$\r\n9 y^{\\prime \\prime}+12 y^{\\prime}+4 y=0, \\quad y(0)=a>0, \\quad y^{\\prime}(0)=-1\r\n$$\r\nFind the critical value of $a$ that separates solutions that become negative from those that are always positive.", "answer_latex": " 1.5", "answer_number": "1.5", "unit": " ", "source": "diff", "problemid": " page172-18", "comment": " ", "solution": ""}]