Can a Substitution Solve a Linear 2nd Order ODE with Non-Constant Coefficients?

[kyin01]

Homework Statement

$$y''x^{2}$$ + 4xy' -y = ln(x)

The Attempt at a Solution

-I considered the quadratic characteristic equation, but it won't work because of the x^2
-I also tried variation of parameters.
so i have v = y' and v'=y'' but i have no idea what i would sub when I get to y.

any ideas?

 

[tiny-tim]

Hi kyin01!

Try changing x …

look for a u(x) such that the equation becomes something simple in dy/du and d²y/du².

 

[yungman]

Homework Statement

$$y''x^{2}$$ + 4xy' -y = ln(x)

The Attempt at a Solution

-I considered the quadratic characteristic equation, but it won't work because of the x^2
-I also tried variation of parameters.
so i have v = y' and v'=y'' but i have no idea what i would sub when I get to y.

any ideas?

I think the standard way is to first find the solutions of the assoc. homogeneous DE $$y''x^{2}$$ + 4xy' -y =0 using Cauchy Euler DE by letting $$y=x^{m}$$.

Then use variation of parameter etc. to find the particular solution.

 

[HallsofIvy]

Yes, although that can make the calculations pretty complicated. As tiny-time suggested, the substitution u= ln(x) will change any "Euler-type" or "Equipotential" equation to a very simple equation with constant coefficients.

I rather suspect that if kyin01 were to check his recent notes or textbook where this problem is given, he would find a discussion of "Euler-type" or "Equipotential" equations. (The name varies from textbook to textbook.)