Internal problem ID [15413]
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: 644.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {\left (x -1\right ) y^{\prime \prime }-x y^{\prime }+y=\left (x -1\right )^{2} {\mathrm e}^{x}} \] Given that one solution of the ode is \begin {align*} y_1 &= {\mathrm e}^{x} \end {align*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 22
dsolve([(x-1)*diff(y(x),x$2)-x*diff(y(x),x)+y(x)=(x-1)^2*exp(x),exp(x)],singsol=all)
\[ y \left (x \right ) = \frac {\left (x^{2}+2 c_{1} -2 x \right ) {\mathrm e}^{x}}{2}+c_{2} x \]
✓ Solution by Mathematica
Time used: 0.067 (sec). Leaf size: 28
DSolve[(x-1)*y''[x]-x*y'[x]+y[x]==(x-1)^2*Exp[x],y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to e^x \left (\frac {x^2}{2}-x+c_1\right )-c_2 x \]