Thread: [SOLVED] initial value differential equation

(x)dy/dx + 2y = 4x^2
y(1) = 2

Solve the initial value problem. Once I get the variable separated I know how to solve the problem. I just can't figure out how to get the x out of the left side. Please help!!

Hello, mathprincess24!

Solve the initial value problem.

We can't separate the variables . . . is that the only method you know?


Divide by

Integrating factor: .

Multiply by

And we have: .

Integrate: .


Since , we have: .


Therefore: .

Originally Posted by mathprincess24

(x)dy/dx + 2y = 4x^2
y(1) = 2

Solve the initial value problem. Once I get the variable separated I know how to solve the problem. I just can't figure out how to get the x out of the left side. Please help!!

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!

thank you so much. that really helps. we learned the integrating factor. i just have a hard time sometimes recognizing when to use it if the problem isnt in the exact general form. thanks again!!