12.9.22 problem 22

Internal problem ID [1778]
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 : 22
Date solved : Monday, January 27, 2025 at 05:34:45 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x y^{\prime \prime }-\left (2 x +2\right ) y^{\prime }+\left (x +2\right ) y&=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+2)*diff(y(x),x)+(x+2)*y(x)=0,exp(x)],singsol=all)
 

\[ y = {\mathrm e}^{x} \left (c_2 \,x^{3}+c_1 \right ) \]

✓ Solution by Mathematica

Time used: 0.023 (sec). Leaf size: 23

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

\[ y(x)\to \frac {1}{3} e^x \left (c_2 x^3+3 c_1\right ) \]