77.1.110 problem 138 (page 198)

Internal problem ID [17929]
Book : V.V. Stepanov, A course of differential equations (in Russian), GIFML. Moscow (1958)
Section : All content
Problem number : 138 (page 198)
Date solved : Monday, March 31, 2025 at 04:51:41 PM
CAS classification : [[_3rd_order, _missing_y]]

\begin{align*} \left (-x^{2}+1\right ) y^{\prime \prime \prime }-x y^{\prime \prime }+y^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.021 (sec). Leaf size: 62

ode:=(-x^2+1)*diff(diff(diff(y(x),x),x),x)-x*diff(diff(y(x),x),x)+diff(y(x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {c_2 \,x^{2} \sqrt {x^{2}-1}-c_3 \,x^{3}+\ln \left (x +\sqrt {x^{2}-1}\right ) \sqrt {x^{2}-1}\, c_3 +c_1 \sqrt {x^{2}-1}+c_3 x}{\sqrt {x^{2}-1}} \]

✓ Mathematica. Time used: 60.085 (sec). Leaf size: 79

ode=(1-x^2)*D[y[x],{x,3}]-x*D[y[x],{x,2}]+D[y[x],x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \int _1^x\left (c_1 \cosh \left (\frac {\arcsin (K[1]) \sqrt {1-K[1]^2}}{\sqrt {K[1]^2-1}}\right )+i c_2 \sinh \left (\frac {\arcsin (K[1]) \sqrt {1-K[1]^2}}{\sqrt {K[1]^2-1}}\right )\right )dK[1]+c_3 \]

✗ Sympy

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

NotImplementedError : The given ODE -x**2*Derivative(y(x), (x, 3)) - x*Derivative(y(x), (x, 2)) + Derivative(y(x), x) + Derivative(y(x), (x, 3)) cannot be solved by the factorable group method