ODE: Combining Undetermined Coeff. & VOP Method

[gabriels-horn]

Title should read "Combining", is there anyway a moderator could alter that so the search function isn't messed up?

Homework Statement

The Attempt at a Solution

I am familiar with both methods, however combining the two is foreign to me. Anyone have any suggestions for this ODE? My guess would be to use the VOP method for the (1/x)*e^x portion and the undetermined coefficients for the 4x^2-3 portion. Any pointers?

 

[LCKurtz]

You have the right idea. I will call the left side of your equation L(y) for brevity.

Your complementary solution y_c satisfies the homogeneous equation

L(y) = 0. Now if you have two particular solutions satisfying L(y_p1) = f(x) and L(y_p2) = g(x), then

L(y_p1+y_p2) = L(y_p1)+L(y_p1) = f(x)+g(x)

so find the two particular solutions separately as you have indicated, and add them to your y_c.

 

[gabriels-horn]

You have the right idea. I will call the left side of your equation L(y) for brevity.

Your complementary solution y_c satisfies the homogeneous equation

L(y) = 0. Now if you have two particular solutions satisfying L(y_p1) = f(x) and L(y_p2) = g(x), then

L(y_p1+y_p2) = L(y_p1)+L(y_p1) = f(x)+g(x)

so find the two particular solutions separately as you have indicated, and add them to your y_c.

Great, thanks for the insight.