Internal problem ID [1114]
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: 8.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
\[ \boxed {y^{\prime \prime }+4 y^{\prime } x +\left (4 x^{2}+2\right ) y=8 \,{\mathrm e}^{-x \left (2+x \right )}} \] Given that one solution of the ode is \begin {align*} y_1 &= {\mathrm e}^{-x^{2}} \end {align*}
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 26
dsolve([diff(y(x),x$2)+4*x*diff(y(x),x)+(4*x^2+2)*y(x)=8*exp(-x*(x+2)),exp(-x^2)],singsol=all)
\[ y \left (x \right ) = \left (c_{1} x +c_{2} \right ) {\mathrm e}^{-x^{2}}+2 \,{\mathrm e}^{-x \left (2+x \right )} \]
✓ Solution by Mathematica
Time used: 0.056 (sec). Leaf size: 29
DSolve[y''[x]+4*x*y'[x]+(4*x^2+2)*y[x]==8*Exp[-x*(x+2)],y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to e^{-x (x+2)} \left (2+e^{2 x} (c_2 x+c_1)\right ) \]