Find the values of a and b so that the general solution of...

Find the values of $a$ and $b$ so that the general solution of the differential equation $y'' + ay' + by = 0$ is given by $y(x) = c_1e^{4x} + c_2e^{-5x}$ (a) What is the value of $a$? (b) What is the value of $b$?

Transcript

00:01 In this question we need to find the values of a and b so that the general solution of the differential equation which is given to us as y double dash plus ay dash plus by equals to 0.

00:29 Its solution is being given as yx is equals to c1 e -d -res to the power 4x plus c2 e -d -a -d -to -the -power negative 5x so here we need to find the values of a and b that is the coefficients of y -dash and coefficient of y so we'll start with the solution of it so we have we have yx to be equals to c1 e to the power 4x plus c2 e to the power negative 5x so now if we take the derivative of it that is y -dash x it will be equals to c1 e -d is to the power 4x here we have 4 plus c2 e -d x to the power negative 5x here we have negative 5x here we have negative 5 now it can be written as 4c1 e raised to the power 4x negative 5 c2 e to the power negative 5x now if we take the derivative of y -dash we'll get by double dash x here we get 16 c1 e -raise to the power 4x positive 25 c2 e as to the power negative 5x now consider a y -dash that is on multiplying y -dash by a we get a 4 c1 e -d to the power 4x negative 5a c2 e -d as to the power negative 5x c2 e -d as to the power negative 5x now on multiplying y by b we get b c1 e raise to the power 4x here we have positive b c2 e raise to the power negative 5x now we have the given differential equation as y double dash plus a y dash plus b y which will be equals to 16 c1 so now now we are now just going to substitute the evaluated y double dash, y dash which is being multiplied to a and by here and we get plus 4a c1 plus b c1 e raised to the power 4x plus 25 c2 negative 5a c2 plus b c2 plus b c2.

03:57 And here we have e -d -to -power -negative 5x.

04:02 Now this term is whole equals to 0.

04:06 So now on solving it further, we get, now we equate the coefficient of e -d -to -the -power -4x and e -d -to -the -power -negative 5x.

04:23 So we get 60c1 plus 4a -c -1 plus b -c -1 to v -0.

04:34 Here we get 25 c2 negative 5a c2 plus b c2 which will be equals to 0...

Question

Find the values of $a$ and $b$ so that the general solution of the differential equation $y'' + ay' + by = 0$ is given by $y(x) = c_1e^{4x} + c_2e^{-5x}$ (a) What is the value of $a$? (b) What is the value of $b$?

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