12.9.12 problem 12

Internal problem ID [1768]
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 : 12
Date solved : Monday, January 27, 2025 at 05:34:37 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} \left (1-2 x \right ) y^{\prime \prime }+2 y^{\prime }+\left (2 x -3\right ) y&=\left (4 x^{2}-4 x +1\right ) {\mathrm e}^{x} \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: 26

dsolve([(1-2*x)*diff(y(x),x$2)+2*diff(y(x),x)+(2*x-3)*y(x)=(1-4*x+4*x^2)*exp(x),exp(x)],singsol=all)
 

\[ y = c_1 x \,{\mathrm e}^{-x}-\frac {{\mathrm e}^{x} \left (x^{2}-2 c_2 -x \right )}{2} \]

✓ Solution by Mathematica

Time used: 0.235 (sec). Leaf size: 77

DSolve[(1-2*x)*D[y[x],{x,2}]+2*D[y[x],x]+(2*x-3)*y[x]==(1-4*x+4*x^2)*Exp[x],y[x],x,IncludeSingularSolutions -> True]
 

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