Solve the following differential equation X(dy/dx)+2y=(x^2)logx?
Set up for integrating factor:
The integrating factor:
Multiply each side by integrating factor:
Integrating the RHS:
Comparing above equation with the standard form of linear D.E.
Integration factor (I,F.)
hence, the complete solution of linear D.E. is given as follows
First, we are going to divide the entire equation by
Now, we need to find the special integrating factor. For a differential equation in this form, the special integrating factor is given by
So we may now multiply the original equation by
If this technique is new to you, the reason that we've multiplied by the special integrating factor is to form the result of a derivative of a product on the left-hand side.
Now, we are almost done. To find
I've just written an antiderivative for the orange integral below, but I'll show you the work for it at the very end.
Integrate by substitution. Let
Perform integration by parts.