75.20.22 problem 661

Internal problem ID [17156]
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 : 661
Date solved : Tuesday, January 28, 2025 at 09:54:42 AM
CAS classification : [[_3rd_order, _missing_y]]

\begin{align*} y^{\prime \prime \prime }+y^{\prime \prime }&=\frac {x -1}{x^{3}} \end{align*}

✓ Solution by Maple

Time used: 0.001 (sec). Leaf size: 45

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

\[ y = -\frac {\left (\int \left (\int \frac {{\mathrm e}^{-x} x^{2} \operatorname {Ei}_{1}\left (-x \right )-2 \,{\mathrm e}^{-x} c_{1} x^{2}+x -1}{x^{2}}d x \right )d x \right )}{2}+c_{2} x +c_{3} \]

✓ Solution by Mathematica

Time used: 16.539 (sec). Leaf size: 58

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

\[ y(x)\to \int _1^x\int _1^{K[3]}e^{-K[2]} \left (c_1+\int _1^{K[2]}\frac {e^{K[1]} (K[1]-1)}{K[1]^3}dK[1]\right )dK[2]dK[3]+c_3 x+c_2 \]