75.20.31 problem 670

Internal problem ID [17165]
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 ODEs). Section 15.5 Linear equations with variable coefficients. The Lagrange method. Exercises page 148
Problem number : 670
Date solved : Tuesday, January 28, 2025 at 08:27:27 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (1-x \right ) y^{\prime \prime }+x y^{\prime }-y&=\left (x -1\right )^{2} {\mathrm e}^{x} \end{align*}

With initial conditions

\begin{align*} y \left (-\infty \right )&=0\\ y^{\prime }\left (0\right )&=1 \end{align*}

✓ Solution by Maple

Time used: 0.278 (sec). Leaf size: 12

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)
 

\[ y = -\frac {x \left (x -2\right ) {\mathrm e}^{x}}{2} \]

✓ Solution by Mathematica

Time used: 0.115 (sec). Leaf size: 16

DSolve[{(1-x)*D[y[x],{x,2}]+x*D[y[x],x]-y[x]==(x-1)^2*Exp[x],{y[-Infinity]==0,Derivative[1][y][0] ==1}},y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to -\frac {1}{2} e^x (x-2) x \]