Answer to Question #124019 in Calculus for jim

Question

Verify that y = e^2x sinx is a solution to the differential equation d^2 y/dx^2 − 4dy/dx + 5y = 0.

Answer 1

Recall: y(x) is said to be solution of differential equation if y(x) satisfied it.

Given differential equation is : y"-4y'+5y=0 ...(1)

Now we have to show that y(x)= e^2xsinx is solution of (1)

Now differentiate y(x) with respect to x ,we get

y'(x)=e^2xcosx+2e^2xsinx

Now differentiate y'(x) with respect to x,we get

y"(x)= 4e^2xcosx+3e^2xsinx

Now put the value of y(x) , y'(x) ,y"(x) into given differential equation (1)

4e^2xcosx+3e^2xsinx-4(e^2xcosx+2e^2xsinx)+5(e^2xsinx)=0

So LHS : 4e^2xcosx+3e^2xsinx-4e^2xcosx-8e^2xsinx+5e^2xsinx

=0 RHS

Hence , y(x) satisfies the given differential equation.

So, y(x) is a solution of the differential equation.