12.9.1 problem 1

Internal problem ID [1757]
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 : 1
Date solved : Monday, January 27, 2025 at 05:34:28 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (2 x +1\right ) y^{\prime \prime }-2 y^{\prime }-\left (2 x +3\right ) y&=\left (2 x +1\right )^{2} \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: 0.008 (sec). Leaf size: 20

dsolve([(2*x+1)*diff(y(x),x$2)-2*diff(y(x),x)-(2*x+3)*y(x)=(2*x+1)^2,exp(-x)],singsol=all)
 

\[ y = c_2 \,{\mathrm e}^{-x}+x \,{\mathrm e}^{x} c_1 -2 x +1 \]

✓ Solution by Mathematica

Time used: 0.131 (sec). Leaf size: 33

DSolve[(2*x+1)*D[y[x],{x,2}]-2*D[y[x],x]-(2*x+3)*y[x]==(2*x+1)^2,y[x],x,IncludeSingularSolutions -> True]
 

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