Internal problem ID [1118]
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: 12.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
Solve \begin {gather*} \boxed {\left (1-2 x \right ) y^{\prime \prime }+2 y^{\prime }+\left (-3+2 x \right ) y-\left (4 x^{2}-4 x +1\right ) {\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: 0.011 (sec). Leaf size: 24
dsolve([(1-2*x)*diff(y(x),x$2)+2*diff(y(x),x)+(2*x-3)*y(x)=(1-4*x+4*x^2)*exp(x),exp(x)],y(x), singsol=all)
\[ y \relax (x ) = c_{2} {\mathrm e}^{x}+c_{1} x \,{\mathrm e}^{-x}-\frac {x \left (x -1\right ) {\mathrm e}^{x}}{2} \]
✓ Solution by Mathematica
Time used: 0.093 (sec). Leaf size: 75
DSolve[(1-2*x)*y''[x]+2*y'[x]+(2*x-3)*y[x]==(1-4*x+4*x^2)*Exp[x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {1}{2} e^x (x-1) x+\frac {c_1 e^{x-\frac {1}{2}} \sqrt {1-2 x}}{\sqrt {2 x-1}}+\frac {c_2 \sqrt {e-2 e x} x (\sinh (x)-\cosh (x))}{\sqrt {2 x-1}} \\ \end{align*}