Internal problem ID [1134]
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: 28.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {\left (1+2 x \right ) x y^{\prime \prime }-2 \left (2 x^{2}-1\right ) y^{\prime }-4 \left (x +1\right ) y=0} \end {gather*} Given that one solution of the ode is \begin {align*} y_1 &= \frac {1}{x} \end {align*}
✓ Solution by Maple
Time used: 0.008 (sec). Leaf size: 16
dsolve([(2*x+1)*x*diff(y(x),x$2)-2*(2*x^2-1)*diff(y(x),x)-4*(x+1)*y(x)=0,1/x],y(x), singsol=all)
\[ y \relax (x ) = \frac {c_{1}}{x}+c_{2} {\mathrm e}^{2 x} \]
✓ Solution by Mathematica
Time used: 0.021 (sec). Leaf size: 28
DSolve[(2*x+1)*x*y''[x]-2*(2*x^2-1)*y'[x]-4*(x+1)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {c_2 e^{2 x+1} x+c_1}{\sqrt {e} x} \\ \end{align*}