Internal problem ID [5587]
Book: Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz.
McGraw-Hill NY. 2007. 1st Edition.
Section: Chapter 2. Second-Order Linear Equations. Section 2.4. THE USE OF A KNOWN
SOLUTION TO FIND ANOTHER. Page 74
Problem number: 4.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [_Gegenbauer]
\[ \boxed {\left (-x^{2}+1\right ) y^{\prime \prime }-2 y^{\prime } x +2 y=0} \] Given that one solution of the ode is \begin {align*} y_1 &= x \end {align*}
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 26
dsolve([(1-x^2)*diff(y(x),x$2)-2*x*diff(y(x),x)+2*y(x)=0,x],y(x), singsol=all)
\[ y \left (x \right ) = c_{1} x +c_{2} \left (\frac {\ln \left (x -1\right ) x}{2}-\frac {\ln \left (x +1\right ) x}{2}+1\right ) \]
✓ Solution by Mathematica
Time used: 0.005 (sec). Leaf size: 19
DSolve[(1-x^2)*y''[x]-2*x*y'[x]+2*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_2 (x \text {arctanh}(x)-1)+c_1 x \\ \end{align*}