75.20.22 problem 661

Internal problem ID [17085]
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 : Monday, March 31, 2025 at 03:40:20 PM
CAS classification : [[_3rd_order, _missing_y]]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 45

ode:=diff(diff(diff(y(x),x),x),x)+diff(diff(y(x),x),x) = (x-1)/x^3; 
dsolve(ode,y(x), singsol=all);
 

\[ y = -\frac {\int \int \frac {{\mathrm e}^{-x} \operatorname {Ei}_{1}\left (-x \right ) x^{2}-2 \,{\mathrm e}^{-x} c_1 \,x^{2}+x -1}{x^{2}}d x d x}{2}+c_2 x +c_3 \]

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

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

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

✓ Sympy. Time used: 0.681 (sec). Leaf size: 44

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(Derivative(y(x), (x, 2)) + Derivative(y(x), (x, 3)) - (x - 1)/x**3,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = C_{1} + C_{2} x + C_{3} e^{- x} - \log {\left (x \right )} + \frac {\frac {1}{2} - e^{- x} \operatorname {E}_{2}\left (- x\right )}{x} + \frac {- \frac {1}{2} + e^{- x} \operatorname {E}_{3}\left (- x\right )}{x^{2}} \]