3.298 problem 1299

Internal problem ID [8878]

Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1299.
ODE order: 2.
ODE degree: 1.

CAS Maple gives this as type [[_2nd_order, _missing_y]]

Solve \begin {gather*} \boxed {\left (a^{2} x^{2}-1\right ) y^{\prime \prime }+2 a^{2} x y^{\prime }=0} \end {gather*}

Solution by Maple

Time used: 0.0 (sec). Leaf size: 31

dsolve((a^2*x^2-1)*diff(diff(y(x),x),x)+2*a^2*x*diff(y(x),x)=0,y(x), singsol=all)
 

\[ y \relax (x ) = c_{1}+\left (\frac {\ln \left (a x -1\right )}{2 a}-\frac {\ln \left (a x +1\right )}{2 a}\right ) c_{2} \]

Solution by Mathematica

Time used: 0.019 (sec). Leaf size: 19

DSolve[2*a^2*x*y'[x] + (-1 + a^2*x^2)*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to c_2-\frac {c_1 \tanh ^{-1}(a x)}{a} \\ \end{align*}