What is it and how to solve it?

Question

Mathematics

It seems like this Second Order Linear Differential Equation is homogeneous, but... what to do with the free term on the right side? and a game on x what to do with it? those. how to decide?

Answer 1

We replace the function t = y/x , as well as the variable with z = -2x , after conjuring, we get
zt'' + (2-z)t' - t/2 = -3/2 (where differentiation is implied by z)
We have given equation to the canonical form for the degenerate hypergeometric equation . Due to the right side, it is non-uniform. But as you remember, the general solution of a linear non-homogeneous equation is the sum of the general solution of a homogeneous equation and a particular solution of a non-homogeneous equation.
For the general solution of a homogeneous degenerate hypergeometric equation, see for example here https://dic.academic.ru/dic.nsf/enc_mathematics/92... . Of course, it is given through the appropriate special functions.F(...) .
Next, we need to find a particular solution to the inhomogeneous equation. It is chosen in an obvious way t(z) = 3 .
Putting it all together, we get the final solution:
y(x) = C1 * x * F( 1/2, 2; -2x ) + C2 * F( -1/2, 0; -2x ) + 3x
Substitute the initial conditions and find the integration constants C1 and C2 are already themselves.

Answer 2

Use either Maple or maybe Mathematica. But if you are still required to explain the decision, I do not envy you. Or numerically.