Internal problem ID [6346]
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: 7.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {y^{\prime \prime }-x f \left (x \right ) y^{\prime }+f \left (x \right ) y=0} \]
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 28
dsolve(diff(y(x),x$2)-x*f(x)*diff(y(x),x)+f(x)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} \left (\int {\mathrm e}^{\int \frac {-2+f \left (x \right ) x^{2}}{x}d x}d x \right ) x +x c_{2} \]
✓ Solution by Mathematica
Time used: 0.287 (sec). Leaf size: 44
DSolve[y''[x]-x*f[x]*y'[x]+f[x]*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to x \left (c_2 \int _1^x\frac {\exp \left (-\int _1^{K[2]}-f(K[1]) K[1]dK[1]\right )}{K[2]^2}dK[2]+c_1\right ) \]