78.11.10 problem 8

Internal problem ID [18281]
Book : DIFFERENTIAL EQUATIONS WITH APPLICATIONS AND HISTORICAL NOTES by George F. Simmons. 3rd edition. 2017. CRC press, Boca Raton FL.
Section : Chapter 3. Second order linear equations. Section 16. The Use of a Known Solution to find Another. Problems at page 121
Problem number : 8
Date solved : Tuesday, January 28, 2025 at 11:45:05 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }-x f \left (x \right ) y^{\prime }+f \left (x \right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.012 (sec). Leaf size: 27

dsolve(diff(y(x),x$2)-x*f(x)*diff(y(x),x)+f(x)*y(x)=0,y(x), singsol=all)
 

\[ y = x \left (c_{1} \left (\int {\mathrm e}^{\int \frac {-2+f \left (x \right ) x^{2}}{x}d x}d x \right )+c_{2} \right ) \]

✓ Solution by Mathematica

Time used: 0.226 (sec). Leaf size: 44

DSolve[D[y[x],{x,2}] -x*f[x]*D[y[x],x]+f[x]*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to x \left (c_2 \int _1^x\frac {\exp \left (-\int _1^{K[2]}-f(K[1]) K[1]dK[1]\right )}{K[2]^2}dK[2]+c_1\right ) \]