(x^2-1)y +xy'-y=0 Don’t know how to solve. Which books are nice discussing about this

[Ask4material]

Hi, everyone. Don’t know how to solve (x^2-1)y"+xy'-y=0

Which books are nice discussing about this kind of differential equations?

Many many thanks


A1. $y\prime=\frac{4x^2}{y}+\frac{y}{x}$

Ans: $y=2x \sqrt{2x+3}$


A2. $y\prime=\frac{2y}{x}-\frac{x^4}{2y}$

Ans: $y=x^2 \sqrt{1-x}$


B1. $(1+x^2)y\prime\prime+3x y\prime =0$

Ans: $y=\frac{x}{\sqrt{1+x^2}}$


B2. $(x^2-1)y\prime\prime +x y\prime-y=0$

Ans: $y=x+ \sqrt{x^2-1}$


B3. $y\prime\prime +\frac{2}{x} y\prime +y=0$

Ans: $y=\frac{\cos{x}}{x}$

 

[HallsofIvy]

Just about any introductory Differential Equations text will have at least one chapter on "linear equations with variable coefficients". The most common method of solution of such equations, however, is to look for a power series solution which might, in some cases, reduce to the formulas you give.

I would also point out that NONE of the "Ans" you give are the general solutions- they are just specific functions, out of the infinite number of solutions, that do sastisfy the equations.

Looking again, I note that A1 and A2 are NOT 'linear'. Such equations are exceptionally difficult and being able to find any "general" solution would be unusual. Also B2 and B3 have "regular singular points" and so a generalization of the power series method, "Frobenius's method", would be used. They, at least, would be treated in any introductory text on Differential equations. Methods for numerical solution and/or determining properties of non-linear equations might be dealt with in more advanced D.E. texts- "Ordinary Differential Equations" by Coddington and Levinson, for example.

 

[grep6]

Nonlinear equations are often solved by making a substitution that puts the equation in a linear form. For the first problem, try multiplying through by y and making the substitution v=y^2 to yield

$x v'(x) - 2v(x) = 8x^3$

then solve, and substitute y back in. Looks like the second problem will be similar.

For more problems/info on this, search for the Bernoulli Differential Equation.
It can be found in some (slightly more advanced) introductory diff-eq books.