Internal problem ID [15435]
Book: A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV,
G.I. MARKARENKO. MIR, MOSCOW. 1983
Section: Chapter 2 (Higher order ODE’s). Section 15.5 Linear equations with variable coefficients.
The Lagrange method. Exercises page 148
Problem number: 670.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {\left (1-x \right ) y^{\prime \prime }+x y^{\prime }-y=\left (x -1\right )^{2} {\mathrm e}^{x}} \] With initial conditions \begin {align*} [y \left (-\infty \right ) = 0, y^{\prime }\left (0\right ) = 1] \end {align*}
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 16
dsolve([(1-x)*diff(y(x),x$2)+x*diff(y(x),x)-y(x)=(x-1)^2*exp(x),y(-infinity) = 0, D(y)(0) = 1],y(x), singsol=all)
\[ y \left (x \right ) = -\frac {x \left (x -2\right ) {\mathrm e}^{x}}{2} \]
✓ Solution by Mathematica
Time used: 0.083 (sec). Leaf size: 16
DSolve[{(1-x)*y''[x]+x*y'[x]-y[x]==(x-1)^2*Exp[x],{y[-Infinity]==0,y'[0]==1}},y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to -\frac {1}{2} e^x (x-2) x \]