How to solve this question 4 Suppose y Ser and y e nonhomoge

How to solve this question?

4. Suppose y Se-r and y e nonhomogeneous equation -6 are both solutions of a certain (a) Isye lor\" + 18e- +6r also a solution of the equation? (b) Is y =-6f also a solution of the equation? (c) What is the general solution of the equation? (d) Determine pi), g), and glt) in the equation above.

Solution

a)

In the two given solutions coefficient of t^2 remains unchanged so that is the particular solution

and the two exponential components of y1 and y2 are solutions to the homogeneous equations ie

y\'\'+py\'+qy=0

a)

No this is not a solution

Coeffficient of particular solution remains unchanged. Only the coefficients of the exponetial terms ie solution to homgeneous equation can change

b) Yes. It is a particular solution

c)

General solution is

y=A e^{2t}+B e^{-5t}-6t^2

d)

First we set up the homogeneous equation

e^{2t} amd e^{-5t} are solutions so characterisitc is

(k-2)(k+5)=k^2+3k-10

Hence homogeneous ode is

y\'\'+3y\'-10=0

So the nonhomogeneous ode is

y\'\'+3y\'-10y=g(t)

-6t^2 is a solution so we substitute and then compute g(t)

-(6t^2)\'\'-3(6t^2)\'+10*6t^2=g(t)

-12-36t+60t^2=g(t)

y\'\'+3y\'-10y=60t^2-36t-12

p(t)=3, q(t)=-10,g(t)=60t^2-36t-12