Find the solution of the given initial value problem: y' - y= 6e^2t. y(0) = 1.

Question

Find the solution of the given initial value problem:     y' - y= 6e^2t. y(0) = 1.

tiffany_rhodes · Answer

I figured you would use the product rule in reverse

anonymous · Answer

$$y'-y=6e^{2t}$$
Find your integrating factor first:
$$\mu(t)=\exp\left(\int (-1)~dt\right)=e^{-t}$$
Multiply both sides of the eq. by $\mu$:
$$e^{-t}y'-e^{-t}y=6e^{t}$$
As you mentioned, the left side is the result of differentiation via product rule:
$$\frac{d}{dt}\left[e^{-t}y\right]=6e^{t}$$
Anti-differentiate both sides:
$$e^{-t}y=6\int e^{t}~dt$$
And so on. You'll have a constant of integration on the right side; plug in the initial conditions to solve for it.

tiffany_rhodes · Answer

thanks @SithsAndGiggles !

anonymous · Answer

yw