12.9.30 problem 30

Internal problem ID [1786]
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 : 30
Date solved : Monday, January 27, 2025 at 05:34:50 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x y^{\prime \prime }-\left (1+4 x \right ) y^{\prime }+\left (4 x +2\right ) y&=0 \end{align*}

Using reduction of order method given that one solution is

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

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 16

dsolve([x*diff(y(x),x$2)-(4*x+1)*diff(y(x),x)+(4*x+2)*y(x)=0,exp(2*x)],singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.025 (sec). Leaf size: 25

DSolve[x*D[y[x],{x,2}]-(4*x+1)*D[y[x],x]+(4*x+2)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

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