Internal problem ID [1120]
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: 14.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
Solve \begin {gather*} \boxed {2 x y^{\prime \prime }+\left (4 x +1\right ) y^{\prime }+\left (1+2 x \right ) y-3 \sqrt {x}\, {\mathrm e}^{-x}=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: 28
dsolve([2*x*diff(y(x),x$2)+(4*x+1)*diff(y(x),x)+(2*x+1)*y(x)=3*x^(1/2)*exp(-x),exp(-x)],y(x), singsol=all)
\[ y \relax (x ) = {\mathrm e}^{-x} c_{2}+\sqrt {x}\, {\mathrm e}^{-x} c_{1}+x^{\frac {3}{2}} {\mathrm e}^{-x} \]
✓ Solution by Mathematica
Time used: 0.02 (sec). Leaf size: 26
DSolve[2*x*y''[x]+(4*x+1)*y'[x]+(2*x+1)*y[x]==3*x^(1/2)*Exp[-x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to e^{-x} \left (\sqrt {x} (x+2 c_2)+c_1\right ) \\ \end{align*}