Hello, mathprincess24!
We can't separate the variables . . . is that the only method you know?
Solve the initial value problem.
Divide by
Integrating factor: .
Multiply by
And we have: .
Integrate: .
Since , we have: .
Therefore: .
You can't. That's not a "separable" equation. It is, however, a linear equation. If you divide by x, you have dy/dx+ 2y/x= 4x. Now you can find an "integrating factor", a function m(x), such that multiplying by it makes the left side a "complete" derivative: d(my)/dx= m dy/dx + (dm/dx)y which much be equal to m dy/dx+ (2/x)y. That means we must have dm/dx= m(2/x) which IS a separable equation: dm/m= 2dx/x which integrates to ln(m)= 2 ln(x)= ln(x^2) or y= x^2 (you can ignore the "constant of integration" because we just want one possible solution). If you multiply the entire equation by x^2 we get x^2 dy/dx+ 2xy= d(x^2y)/dx= 4x^3 which we can now write as d(x^2y)= 4x^4 dx and integrate to get x^2y= x^4+ C. Set x= 1, y= 2 to find C.
Once again Soroban beat me! And now I have been beaten by AIR by a nose!