50.12.10 problem 7

Internal problem ID [8014]
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
Date solved : Monday, January 27, 2025 at 03:36:49 PM
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: 0.027 (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)
 

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

✓ Solution by Mathematica

Time used: 0.216 (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]
 

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