Verify that the function y satisfies the given differential equation: y" - 6y' + 5y =5.?

Question

Verify that the function y satisfies the given differential equation: y" - 6y' + 5y =5.
y = e^5x - 4e^x +1.

Answer 1

y' = 5e^5x - 4e^x
y'' = 25e^5x - 4e^x

y'' - 6y' + 5y = 5
(25e^5x - 4e^x) - 6(5e^5x - 4e^x) + 5(e^5x - 4e^x +1) = 5
25e^5x - 4e^x - 30e^5x + 24e^x + 5e^5x - 20e^x + 5 = 5
(25 - 30 + 5)e^5x + (-4 + 24 - 20)e^x + 5 = 5
0e^5x + 0e^x + 5 = 5
5 = 5

Answer 2

First find the first and second derivatives of y:
y' = 5*e^(5*x) - 4e^x
y'' = 25e^(5x)-4e^x.

Now substitute to verify:
25e^(5x)-4e^x -6(5*e^(5*x) - 4e^x)+5(e^(5x)-4e^x+1) = 5
Simplify to make sure the right side equals the left carefully using the distributive property.

Answer 3

y = e^5x - 4e^x +1.
y' = 5e^5x - 4e^x
y'' = 25e^5x - 4e^x

fill this in into

y" - 6y' + 5y =5

if the identity =5 holds then its ok otherwise you or i or the teacher made a calculation error.

.

Answer 4

Take the first derivative of y.

Find the second derivative of y.

Plug y", y', and y into the differential equation.