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/* ODE tests - Second order linear equations 1-100 Reference: E Kamke, Differentialgleichungen Losungsmethoden und Losungen, Vol 1, Geest & Portig, Leipzig, 1961 */ /* Print ode number*/ (pn_(n_):=print("Kamke ODE 2.",n_),true); true; /* Kamke 2. 1 */ (pn_(1),ans:contrib_ode(eqn:'diff(y,x,2),y,x)); [y=%k2*x+%k1]; ode_check(eqn,ans[1]); 0; method; constcoeff; /* Kamke 2. 2 */ (pn_(2),ans:contrib_ode(eqn:'diff(y,x,2)+y,y,x)); [y=%k1*sin(x)+%k2*cos(x)]; ode_check(eqn,ans[1]); 0; method; constcoeff; /* Kamke 2. 3 */ (pn_(3),ans:contrib_ode(eqn:'diff(y,x,2)+y-sin(n*x),y,x)); [y=-sin(n*x)/(n^2-1)+%k1*sin(x)+%k2*cos(x)]; ode_check(eqn,ans[1]); 0; method; variationofparameters; /* Kamke 2. 4 */ (pn_(4),ans:contrib_ode(eqn:'diff(y,x,2)+y-a*cos(b*x),y,x)); [y=-a*cos(b*x)/(b^2-1)+%k1*sin(x)+%k2*cos(x)]; ode_check(eqn,ans[1]); 0; method; variationofparameters; /* Kamke 2. 5 */ (pn_(5),ans:contrib_ode(eqn:'diff(y,x,2)+y-sin(a*x)*sin(b*x),y,x)); [y=((b^2-2*a*b+a^2-1)*cos((b+a)*x)+(-b^2-2*a*b-a^2+1)*cos((b-a)*x))/(2*b^4+(-4*a^2-4)*b^2+2*a^4-4*a^2+2)+%k1*sin(x)+%k2*cos(x)]; [method,ode_check(eqn,ans[1])]; [variationofparameters,0]; /* Kamke 2. 6 */ (pn_(6),ans:contrib_ode(eqn:'diff(y,x,2)-y,y,x)); [y=%k1*%e^x+%k2*%e^-x]; ode_check(eqn,ans[1]); 0; method; constcoeff; /* Kamke 2. 7 */ (pn_(7),ans:contrib_ode(eqn:'diff(y,x,2)-2*y-4*x^2*%e^x^2,y,x)); [y=%e^x^2+%k1*%e^(2^(1/2)*x)+%k2*%e^-(2^(1/2)*x)]; ode_check(eqn,ans[1]); 0; method; variationofparameters; /* Kamke 2. 8 CHECKME */ assume(a>0); [a>0]; (pn_(8),ans:contrib_ode(eqn:'diff(y,x,2)+a^2*y-cot(a*x),y,x)); [y = -((sin(a*x)*log(cos(a*x)+1)-sin(a*x)*log(cos(a*x)-1))/(2*a^2)) +%k1*sin(a*x)+%k2*cos(a*x)]; [method,ode_check(eqn,ans[1])]; [variationofparameters,0]; forget(a>0); [a>0]; /* Kamke 2. 9 */ assume(l>0); [l>0]; (pn_(9),ans:contrib_ode(eqn:'diff(y,x,2)+l*y,y,x)); [y=%k1*sin(sqrt(l)*x)+%k2*cos(sqrt(l)*x)]; ode_check(eqn,ans[1]); 0; method; constcoeff; forget(l>0); [l>0]; /* Kamke 2. 10 */ (pn_(10),ans:contrib_ode(eqn:'diff(y,x,2)+(a*x+b)*y,y,x)); [y = bessel_y(1/3,2*(a*x+b)^(3/2)/(3*abs(a)))*%k2*sqrt(a*x+b) + bessel_j(1/3,2*(a*x+b)^(3/2)/(3*abs(a)))*%k1*sqrt(a*x+b)]; [method,ode_check(eqn,ans[1])]; [odelin,0]; /* Kamke 2. 11 */ (pn_(11),ans:contrib_ode(eqn:'diff(y,x,2)-(x^2+1)*y,y,x)); [y = %k2*%e^(x^2/2)*erf(x)+%k1*%e^(x^2/2)]; [method,ode_check(eqn,ans[1])]; [odelin,0]; /* Kamke 2. 12 */ (pn_(12),ans:contrib_ode(eqn:'diff(y,x,2)-(x^2+a)*y,y,x),0); 0$ /* [y = (%k2*kummer_u(-(a-1)/4,1/2,-x^2)*%e^(x^2/2)*sqrt(abs(x)))/sqrt(-x) + (%k1*kummer_m(-(a-1)/4,1/2,-x^2)*%e^(x^2/2)*sqrt(abs(x)))/sqrt(-x)]; */ [method,ode_check(eqn,ans[1])]; [odelin,0]; /* Kamke 2. 13 */ (pn_(13),ans:contrib_ode(eqn:'diff(y,x,2)-(a^2*x^2+a)*y,y,x)); [y = %k2*%e^(a*x^2/2)*erf(sqrt(a)*x)+%k1*%e^(a*x^2/2)]; [method,ode_check(eqn,ans[1])]; [odelin,0]; /* Kamke 2. 14 */ (pn_(14),ans:contrib_ode(eqn:'diff(y,x,2)-c*x^a*y,y,x)); false; /* Kamke 2. 15 */ (pn_(15),ans:contrib_ode(eqn:'diff(y,x,2)-(a^2*x^(2*n)-1)*y,y,x)); false; /* Kamke 2. 16 */ (pn_(16),ans:contrib_ode(eqn:'diff(y,x,2)+(a*x^(2*c)+b*x^(c-1))*y,y,x)); false; /* Kamke 2. 17 */ (pn_(17),ans:contrib_ode(eqn:'diff(y,x,2)+(%e^(2*x)-v^2)*y,y,x)); false; /* Kamke 2. 18 */ (pn_(18),ans:contrib_ode(eqn:'diff(y,x,2)+a*%e^(b*x)*y,y,x)); false; /* Kamke 2. 19 */ (pn_(19),ans:contrib_ode(eqn:'diff(y,x,2)-(4*a^2*b^2*x^2*%e^(2*b*x^2)-1)*y,y,x)); false; /* Kamke 2. 20 */ (pn_(20),ans:contrib_ode(eqn:'diff(y,x,2)+(a*%e^(2*x)+b*%e^x+c)*y,y,x)); false; /* Kamke 2. 21 */ (pn_(21),ans:contrib_ode(eqn:'diff(y,x,2)+(a*cos(x)^2+b)*y,y,x)); false; /* Kamke 2. 22 */ (pn_(22),ans:contrib_ode(eqn:'diff(y,x,2)+(a*cos(2*x)+b)*y,y,x)); false; /* Kamke 2. 23 */ (pn_(23),ans:contrib_ode(eqn:'diff(y,x,2)+(a*cos(x)^2+b)*y,y,x)); false; /* Kamke 2. 24 */ (pn_(24),ans:contrib_ode(eqn:'diff(y,x,2)-(2*tan(x)^2+1)*y,y,x)); false; /* Kamke 2. 25 */ (pn_(25),ans:contrib_ode(eqn:'diff(y,x,2)-((n-1)*n/sin(x)^2+(m-1)*m/cos(x)^2+a)*y,y,x)); false; /* Kamke 2. 26 */ (pn_(26),ans:contrib_ode(eqn:'diff(y,x,2)-y*(B+p(x)*A),y,x)); false; /* Kamke 2. 27 */ (pn_(27),ans:contrib_ode(eqn:'diff(y,x,2)+(a*jacobi_sn(x,m)^2+b)*y,y,x)); false; /* Kamke 2. 28 */ (pn_(28),ans:contrib_ode(eqn:'diff(y,x,2)-('diff(p(x),x,4)/30+7*'diff(p(x),x,2)/3+a*p(x)+b)*y,y,x)); false; /* Kamke 2. 29 */ (pn_(29),ans:contrib_ode(eqn:'diff(y,x,2)-('diff(f(x),x,1)+f(x)^2)*y,y,x)); false; /* Kamke 2. 30 */ (pn_(30),ans:contrib_ode(eqn:'diff(y,x,2)+(P(x)+l)*y,y,x)); false; /* Kamke 2. 31 */ (pn_(31),ans:contrib_ode(eqn:'diff(y,x,2)-f(x)*y,y,x)); false; /* Kamke 2. 32 */ (pn_(32),ans:contrib_ode(eqn:'diff(y,x,2)+('diff(g(x),x,3)/(2*'diff(g(x),x,1))-3*('diff(g(x),x,2))^2/(4*('diff(g(x),x,1))^2)+(1/4-v^2)*('diff(g(x),x,1))^2/g(x)+('diff(g(x),x,1))^2)*y,y,x)); false; /* Kamke 2. 33 */ (pn_(33),ans:contrib_ode(eqn:'diff(y,x,2)+'diff(y,x,1)+a*%e^-(2*x)*y,y,x)); [y=%k2*cos(sqrt(a)*%e^-x)-%k1*sin(sqrt(a)*%e^-x)]; ode_check(eqn,ans[1]); 0; method; xformtoconstcoeff; /* Kamke 2. 34 */ (pn_(34),ans:contrib_ode(eqn:'diff(y,x,2)-'diff(y,x,1)+%e^(2*x)*y,y,x)); [y=%k1*sin(%e^x)+%k2*cos(%e^x)]; ode_check(eqn,ans[1]); 0; method; xformtoconstcoeff; /* Kamke 2. 35 */ assume(a^2>4*b); [a^2>4*b]; (pn_(35),ans:contrib_ode(eqn:'diff(y,x,2)+a*'diff(y,x,1)+b*y,y,x)); [y=%k1*%e^((sqrt(a^2-4*b)-a)*x/2)+%k2*%e^((-sqrt(a^2-4*b)-a)*x/2)]; ode_check(eqn,ans[1]); 0; method; constcoeff; forget(a^2>4*b); [a^2>4*b]; /* Kamke 2. 36 */ assume(a^2>4*b); [a^2>4*b]; (pn_(36),ans:contrib_ode(eqn:'diff(y,x,2)+a*'diff(y,x,1)+b*y-f(x),y,x),0); /* [y=-%e^(-sqrt(a^2-4*b)*x/2-a*x/2)*('integrate(%e^(sqrt(a^2-4*b)*x/2+a*x/2)*f(x),x)-%e^(sqrt(a^2-4*b)*x)*'integrate(%e^(a*x/2-sqrt(a^2-4*b)*x/2)*f(x),x))/sqrt(a^2-4*b)+%k1*%e^((sqrt(a^2-4*b)-a)*x/2)+%k2*%e^((-sqrt(a^2-4*b)-a)*x/2)]; */ 0$ ode_check(eqn,ans[1]); 0$ method; variationofparameters; forget(a^2>4*b); [a^2>4*b]; /* Kamke 2. 37 */ (pn_(37),ans:contrib_ode(eqn:'diff(y,x,2)+a*'diff(y,x,1)+(-b^2*x^2-c)*y,y,x),0); 0$ /* [y = (%k2*kummer_u(-(4*c-4*b+a^2)/(16*b),1/2,-b*x^2) *%e^((b*x^2)/2-(a*x)/2)*sqrt(abs(x)))/sqrt(-x) +(%k1*kummer_m(-(4*c-4*b+a^2)/(16*b),1/2,-b*x^2) *%e^((b*x^2)/2-(a*x)/2)*sqrt(abs(x)))/sqrt(-x)]$ */ [method,ode_check(eqn,expand(ans[1]))]; [odelin,0]; /* Kamke 2. 38 */ (pn_(38),ans:contrib_ode(eqn:'diff(y,x,2)+2*a*'diff(y,x,1)+f(x)*y,y,x)); false; /* Kamke 2. 39 */ (pn_(39),ans:contrib_ode(eqn:'diff(y,x,2)+x*'diff(y,x,1)+y,y,x)); [y=%k2*%e^-(x^2/2)-sqrt(2)*sqrt(%pi)*%i*%k1*%e^-(x^2/2)*erf(%i*x/sqrt(2))/2]; ode_check(eqn,ans[1]); 0; method; exact; /* Kamke 2. 40 */ (pn_(40),ans:contrib_ode(eqn:'diff(y,x,2)+x*'diff(y,x,1)-y,y,x),0); 0$ /* [y = kummer_u(0,3/2,-x^2/2)*%k2*x+kummer_m(0,3/2,-x^2/2)*%k1*x]; */ [method,ode_check(eqn,ans[1])]; [odelin,0]; /* Kamke 2. 41 */ (pn_(41),ans:contrib_ode(eqn:'diff(y,x,2)+x*'diff(y,x,1)+(n+1)*y,y,x),0); 0$ /* [y = %k2*kummer_u(n/2+1/2,1/2,-(x^2/2)) +%k1*kummer_m(n/2+1/2,1/2,-(x^2/2))]; */ [method,ode_check(eqn,ans[1])]; [odelin,0]; /* Kamke 2. 42 */ (pn_(42),ans:contrib_ode(eqn:'diff(y,x,2)+x*'diff(y,x,1)-n*y,y,x),0); 0$ /* [y = %k2*kummer_u(-n/2,1/2,-x^2/2) +%k1*kummer_m(-n/2,1/2,-x^2/2)]; */ [method,ode_check(eqn,ans[1])]; [odelin,0]; /* Kamke 2. 43 */ (pn_(43),ans:contrib_ode(eqn:'diff(y,x,2)-x*'diff(y,x,1)+2*y,y,x),0); 0$ /* [y = kummer_u(2,3/2,-x^2/2)*%k2*x*%e^(x^2/2) +kummer_m(2,3/2,-x^2/2)*%k1*x*%e^(x^2/2)]; */ [method,ode_check(eqn,ans[1])]; [odelin,0]; /* Kamke 2. 44 */ (pn_(44),ans:contrib_ode(eqn:'diff(y,x,2)-x*'diff(y,x,1)-a*y,y,x),0); 0$ /* [y = %k2*kummer_u(1/2-(a/2),1/2,-(x^2/2))*%e^(x^2/2) +%k1*kummer_m(1/2-(a/2),1/2,-(x^2/2))*%e^(x^2/2)]; */ [method,ode_check(eqn,ans[1])]; [odelin,0]; /* Kamke 2. 45 */ (pn_(45),ans:contrib_ode(eqn:'diff(y,x,2)-x*'diff(y,x,1)+(x-1)*y,y,x),0); 0$ /* [y = %k2*%e^(x-2)*erf(%i*x/sqrt(2)-2*%i/sqrt(2))+%k1*%e^x]; */ [method,ode_check(eqn,ans[1])];