Internal problem ID [15416]
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: 647.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {y^{\prime \prime }-y^{\prime }+{\mathrm e}^{2 x} y=x \,{\mathrm e}^{2 x}-1} \] Given that one solution of the ode is \begin {align*} y_1 &= \sin \left ({\mathrm e}^{x}\right ) \end {align*}
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 16
dsolve([diff(y(x),x$2)-diff(y(x),x)+exp(2*x)*y(x)=x*exp(2*x)-1,sin(exp(x))],singsol=all)
\[ y \left (x \right ) = \sin \left ({\mathrm e}^{x}\right ) c_{2} +\cos \left ({\mathrm e}^{x}\right ) c_{1} +x \]
✓ Solution by Mathematica
Time used: 0.341 (sec). Leaf size: 21
DSolve[y''[x]-y'[x]+Exp[2*x]*y[x]==x*Exp[2*x]-1,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to x+c_1 \cos \left (e^x\right )+c_2 \sin \left (e^x\right ) \]