12.9.3 problem 3

Internal problem ID [1759]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 5 linear second order equations. Section 5.6 Reduction or order. Page 253
Problem number : 3
Date solved : Monday, January 27, 2025 at 05:34:30 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }-x y^{\prime }+y&=x \end{align*}

Using reduction of order method given that one solution is

\begin{align*} y&=x \end{align*}

✓ Solution by Maple

Time used: 0.005 (sec). Leaf size: 18

dsolve([x^2*diff(y(x),x$2)-x*diff(y(x),x)+y(x)=x,x],singsol=all)
 

\[ y = x \left (c_2 +\ln \left (x \right ) c_1 +\frac {\ln \left (x \right )^{2}}{2}\right ) \]

✓ Solution by Mathematica

Time used: 0.019 (sec). Leaf size: 25

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

\[ y(x)\to \frac {1}{2} x \left (\log ^2(x)+2 c_2 \log (x)+2 c_1\right ) \]