Internal problem ID [1136]
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.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {x y^{\prime \prime }-\left (4 x +1\right ) y^{\prime }+\left (4 x +2\right ) y=0} \end {gather*} Given that one solution of the ode is \begin {align*} y_1 &= {\mathrm e}^{2 x} \end {align*}
✓ Solution by Maple
Time used: 0.005 (sec). Leaf size: 20
dsolve([x*diff(y(x),x$2)-(4*x+1)*diff(y(x),x)+(4*x+2)*y(x)=0,exp(2*x)],y(x), singsol=all)
\[ y \relax (x ) = c_{1} {\mathrm e}^{2 x}+c_{2} {\mathrm e}^{2 x} x^{2} \]
✓ Solution by Mathematica
Time used: 0.007 (sec). Leaf size: 25
DSolve[x*y''[x]-(4*x+1)*y'[x]+(4*x+2)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{2} e^{2 x} \left (c_2 x^2+2 c_1\right ) \\ \end{align*}