12.10.30 problem 30

Internal problem ID [1834]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 5 linear second order equations. Section 5.7 Variation of Parameters. Page 262
Problem number : 30
Date solved : Monday, January 27, 2025 at 05:36:49 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} \left (3 x -1\right ) y^{\prime \prime }-\left (3 x +2\right ) y^{\prime }-\left (6 x -8\right ) y&=\left (3 x -1\right )^{2} {\mathrm e}^{2 x} \end{align*}

With initial conditions

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

✓ Solution by Maple

Time used: 0.020 (sec). Leaf size: 28

dsolve([(3*x-1)*diff(y(x),x$2)-(3*x+2)*diff(y(x),x)-(6*x-8)*y(x)=(3*x-1)^2*exp(2*x),y(0) = 1, D(y)(0) = 2],y(x), singsol=all)
 

\[ y = \frac {\left (3 x^{2}-2 x +6\right ) {\mathrm e}^{2 x}}{6}+\frac {x \,{\mathrm e}^{-x}}{3} \]

✓ Solution by Mathematica

Time used: 0.124 (sec). Leaf size: 34

DSolve[{(3*x-1)*D[y[x],{x,2}]-(3*x+2)*D[y[x],x]-(6*x-8)*y[x]==(3*x-1)^2*Exp[2*x],{y[0]==1,Derivative[1][y][0] ==2}},y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {1}{6} e^{-x} \left (e^{3 x} \left (3 x^2-2 x+6\right )+2 x\right ) \]