77.1.90 problem 117 (page 177)

Internal problem ID [17909]
Book : V.V. Stepanov, A course of differential equations (in Russian), GIFML. Moscow (1958)
Section : All content
Problem number : 117 (page 177)
Date solved : Monday, March 31, 2025 at 04:49:50 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.260 (sec). Leaf size: 477

ode:=y(x)^2*(x^2*diff(diff(y(x),x),x)-x*diff(y(x),x)+y(x)) = x^3; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} \text {Solution too large to show}\end{align*}

✓ Mathematica. Time used: 2.634 (sec). Leaf size: 268

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

\begin{align*} \text {Solve}\left [\log (x)-\frac {2 x^{3/2} \sqrt {2-\frac {c_1 y(x)}{x}} \arcsin \left (\frac {\sqrt {c_1} \sqrt {y(x)}}{\sqrt {2} \sqrt {x}}\right )-2 \sqrt {c_1} x \sqrt {y(x)}+c_1{}^{3/2} y(x)^{3/2}}{c_1{}^{3/2} x \sqrt {-\frac {x^2}{y(x)^2}} \sqrt {y(x)} \sqrt {\frac {y(x) (2 x-c_1 y(x))}{x^2}}}&=c_2,y(x)\right ] \\ \text {Solve}\left [\log (x)+\frac {2 x^{3/2} \sqrt {2-\frac {c_1 y(x)}{x}} \arcsin \left (\frac {\sqrt {c_1} \sqrt {y(x)}}{\sqrt {2} \sqrt {x}}\right )-2 \sqrt {c_1} x \sqrt {y(x)}+c_1{}^{3/2} y(x)^{3/2}}{c_1{}^{3/2} x \sqrt {-\frac {x^2}{y(x)^2}} \sqrt {y(x)} \sqrt {\frac {y(x) (2 x-c_1 y(x))}{x^2}}}&=c_2,y(x)\right ] \\ \end{align*}

✗ Sympy

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

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