9.1 problem 1

Internal problem ID [1107]

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: 1.
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 ) y^{\prime \prime }-2 y^{\prime }-\left (2 x +3\right ) y-\left (1+2 x \right )^{2}=0} \end {gather*} Given that one solution of the ode is \begin {align*} y_1 &= {\mathrm e}^{-x} \end {align*}

Solution by Maple

Time used: 0.015 (sec). Leaf size: 20

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

\[ y \relax (x ) = {\mathrm e}^{-x} c_{2}+x \,{\mathrm e}^{x} c_{1}+1-2 x \]

Solution by Mathematica

Time used: 0.06 (sec). Leaf size: 33

DSolve[(2*x+1)*y''[x]-2*y'[x]-(2*x+3)*y[x]==(2*x+1)^2,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to c_1 e^{-x-\frac {1}{2}}+x \left (-2+c_2 e^{x+\frac {1}{2}}\right )+1 \\ \end{align*}