is equal to zero then the first derivative is zero the second derivative is zero all the derivatives are zero blah blah blah therefore our equation is also satisfied by this function in other words we have two solutions to this equation great now this equation did not ask for it but i'd like to give you as an extension to this problem because it could have right what would happen if they gave us initial conditions right i hope you heard about them the initial conditions actually allow you to determine the constants in this case because notice that our solution contains two constants c and d and they're undetermined but if we were given initial conditions such as f of NUMBER_TOKEN equals NUMBER_TOKEN and f prime at NUMBER_TOKEN equals negative one over e we could find the value of c and d very easily okay so you want us to work it out well f of zero equals one basically means that you can just plug in well let's go ahead and write this down first this is our solution one of the solutions of course i'm not talking about NUMBER_TOKEN here by the way because if y is NUMBER_TOKEN then it's going to be always NUMBER_TOKEN for any x value so this is not going to work with y equals NUMBER_TOKEN. but for the other solution that for which y does not equal NUMBER_TOKEN you can just plug in you know x equals zero you're gonna get negative ln c plus d is equal to one and then from the second equation the derivative of this one remember was negative one over x plus c and if you replace x with NUMBER_TOKEN you will get negative NUMBER_TOKEN over c equals negative NUMBER_TOKEN over e from here you're going to get c equals e and if you plug in c equals e here you're going to get negative ln e plus d equals one or d equals two and then by no finding the values of c and d you can basically write the solution as negative ln x plus e plus d and of course d must be replaced with two and that's going to give us the initial condition solution it wasn't given that way that's why our solutions are basically these two and this brings us to the end of this video well thank you for watching i hope you enjoyed it please let me know don't forget to comment like and subscribe i'll see you tomorrow with another video until then be safe take care and bye bye
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