75.8.7 problem 205

Internal problem ID [16752]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Section 8. First order not solved for the derivative. Exercises page 67
Problem number : 205
Date solved : Monday, March 31, 2025 at 03:15:53 PM
CAS classification : [_quadrature]

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

✓ Maple. Time used: 0.028 (sec). Leaf size: 28

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

\begin{align*} y &= -\frac {i x^{2}}{2}+c_1 \\ y &= \frac {i x^{2}}{2}+c_1 \\ y &= c_1 \,{\mathrm e}^{x} \\ \end{align*}

✓ Mathematica. Time used: 0.01 (sec). Leaf size: 43

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

\begin{align*} y(x)\to c_1 e^x \\ y(x)\to c_1-\frac {i x^2}{2} \\ y(x)\to \frac {i x^2}{2}+c_1 \\ \end{align*}

✓ Sympy. Time used: 0.258 (sec). Leaf size: 27

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

\[ \left [ y{\left (x \right )} = C_{1} - \frac {i x^{2}}{2}, \ y{\left (x \right )} = C_{1} + \frac {i x^{2}}{2}, \ y{\left (x \right )} = C_{1} e^{x}\right ] \]