75.20.9 problem 644

Internal problem ID [17143]
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 : 644
Date solved : Tuesday, January 28, 2025 at 09:54:09 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using reduction of order method given that one solution is

\begin{align*} y&={\mathrm e}^{x} \end{align*}

✓ Solution by Maple

Time used: 0.009 (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)
 

\[ y = \frac {\left (x^{2}+2 c_{1} -2 x \right ) {\mathrm e}^{x}}{2}+c_{2} x \]

✓ Solution by Mathematica

Time used: 0.234 (sec). Leaf size: 248

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

\[ y(x)\to \exp \left (\int _1^x\frac {K[1]-2}{2 (K[1]-1)}dK[1]-\frac {1}{2} \int _1^x-\frac {K[2]}{K[2]-1}dK[2]\right ) \left (\int _1^x-\exp \left (K[4]+\int _1^{K[4]}\frac {K[1]-2}{2 (K[1]-1)}dK[1]+\frac {1}{2} \int _1^{K[4]}-\frac {K[2]}{K[2]-1}dK[2]\right ) (K[4]-1) \int _1^{K[4]}\exp \left (-2 \int _1^{K[3]}\frac {K[1]-2}{2 (K[1]-1)}dK[1]\right )dK[3]dK[4]+\int _1^x\exp \left (-2 \int _1^{K[3]}\frac {K[1]-2}{2 (K[1]-1)}dK[1]\right )dK[3] \left (\int _1^x\exp \left (K[5]+\int _1^{K[5]}\frac {K[1]-2}{2 (K[1]-1)}dK[1]+\frac {1}{2} \int _1^{K[5]}-\frac {K[2]}{K[2]-1}dK[2]\right ) (K[5]-1)dK[5]+c_2\right )+c_1\right ) \]