12.9.32 problem 32

Internal problem ID [1788]
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 : 32
Date solved : Monday, January 27, 2025 at 05:34:52 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (3 x -1\right ) y^{\prime \prime }-\left (3 x +2\right ) y^{\prime }-\left (6 x -8\right ) y&=0 \end{align*}

Using reduction of order method given that one solution is

\begin{align*} y&={\mathrm e}^{2 x} \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=2\\ y^{\prime }\left (0\right )&=3 \end{align*}

✓ Solution by Maple

Time used: 0.014 (sec). Leaf size: 18

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

\[ y = 2 \,{\mathrm e}^{2 x}-x \,{\mathrm e}^{-x} \]

✓ Solution by Mathematica

Time used: 0.145 (sec). Leaf size: 21

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

\[ y(x)\to 2 e^{2 x}-e^{-x} x \]