Please help with this differential equation:
x dy/dx - 2y = 2x^4, y(2) = 8
First, I recognized it as a linear diff. eq. and found the integrating factor to be x^2. Then multiplied the whole eq. by this factor and arranged it to get the integral. Then my result came out to be
x^2 y = (x^6)/3 +C, using initial condition given found C to be 32/3. The answer in the book is
y = x^4 - 2x^2 . Please help to figure this out.

Question

Please help with this differential equation: 
x dy/dx - 2y = 2x^4, y(2) = 8
First, I recognized it as a linear diff. eq. and found the integrating factor to be x^2. Then multiplied the whole eq. by this factor and arranged it to get the integral. Then my result came out to be 
x^2 y = (x^6)/3  +C, using initial condition given found C to be 32/3. The answer in the book is 
y = x^4 - 2x^2 . Please help to figure this out.

jamesj · Answer

Writing first the equation in 'standard form', where the coefficient of dy/dx is 1, we have:

$$\frac{dy}{dx} - \frac{2}{x}y = 2x^3$$

Not the integrating factor is the exponential of the integral of the coefficient of y, which is NOT x^2, rather it is ....

Getting this factor right should clear this up for you.

jamesj · Answer

NoW the integrating factor ....

jamesj · Answer

Making sense?