75.20.11 problem 646

Internal problem ID [17145]
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 : 646
Date solved : Tuesday, January 28, 2025 at 09:54:11 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} \left (x^{4}-x^{3}\right ) y^{\prime \prime }+\left (2 x^{3}-2 x^{2}-x \right ) y^{\prime }-y&=\frac {\left (x -1\right )^{2}}{x} \end{align*}

Using reduction of order method given that one solution is

\begin{align*} y&=\frac {1}{x} \end{align*}

✓ Solution by Maple

Time used: 0.009 (sec). Leaf size: 22

dsolve([(x^4-x^3)*diff(y(x),x$2)+(2*x^3-2*x^2-x)*diff(y(x),x)-y(x)=(x-1)^2/x,1/x],singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.404 (sec). Leaf size: 281

DSolve[(x^4-x^3)*D[y[x],{x,2}]+(2*x^3-2*x^2-x)*D[y[x],x]-y[x]==(x-1)^2/x,y[x],x,IncludeSingularSolutions -> True]
 

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