Question

HELP!
Find the solution of the differential equation

(1+x^8)y'+8x^7y=2e^2x , such that y(0)=0

Answer 1

Solve the corresponding homogenous equation first.
\[\Large (1+x^8)y'+8x^7y=0 \]
Rearrange it to get:
\[\Large y'+ \frac{8x^7}{1+x^8}y=0 \]

Answer 2

So your integration factor is pretty obvious here, \[\Large \mu(x)=1+x^8 \] In fact, you can apply this to the general form of the differential equation, it's not necessary to solve the homogenous first.

Answer 3

Rather a matter of art I would say, depending if one rather prefers to use Variation of Constants or partial integration.

Answer 4

Notice that (d/dx)(1 + x^8) = 8x^7 hence you write the left hand side as

\[ \frac{d\ }{dx}[(1+x^8)y] = (1+x^8)y' + 8x^7y\]
and therefore the equation can be written as
\[ \frac{d\ }{dx}[(1+x^8)y] = 2e^{2x} \]

Now integrate both sides

Answer 5

ok thanks!