50.12.11 problem 8

Internal problem ID [8015]
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 : 8
Date solved : Monday, January 27, 2025 at 03:36:50 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x y^{\prime \prime }-\left (2 x +1\right ) y^{\prime }+y \left (1+x \right )&=0 \end{align*}

Using reduction of order method given that one solution is

\begin{align*} y&={\mathrm e}^{x} \end{align*}

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 14

dsolve([x*diff(y(x),x$2)-(2*x+1)*diff(y(x),x)+(x+1)*y(x)=0,exp(x)],singsol=all)
 

\[ y = {\mathrm e}^{x} \left (c_{2} x^{2}+c_{1} \right ) \]

✓ Solution by Mathematica

Time used: 0.027 (sec). Leaf size: 23

DSolve[x*D[y[x],{x,2}]-(2*x+1)*D[y[x],x]+(x+1)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

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