83.23.15 problem 15

Internal problem ID [19217]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Chapter V. Singular solutions. Exercise V at page 76
Problem number : 15
Date solved : Thursday, March 13, 2025 at 01:55:52 PM
CAS classification : [[_1st_order, _with_linear_symmetries]]

\begin{align*} {y^{\prime }}^{2}+2 y^{\prime } x^{3}-4 x^{2} y&=0 \end{align*}

✓ Maple. Time used: 0.142 (sec). Leaf size: 19

ode:=diff(y(x),x)^2+2*diff(y(x),x)*x^3-4*x^2*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y \left (x \right ) &= -\frac {x^{4}}{4} \\ y \left (x \right ) &= c_{1} \left (x^{2}+c_{1} \right ) \\ \end{align*}

✓ Mathematica. Time used: 0.818 (sec). Leaf size: 142

ode=D[y[x],x]^2+2*D[y[x],x]*x^3-4*x^2*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} \text {Solve}\left [\frac {1}{4} \log (y(x))-\frac {\sqrt {x^6+4 x^2 y(x)} \text {arctanh}\left (\frac {\sqrt {x^4+4 y(x)}}{x^2}\right )}{2 x \sqrt {x^4+4 y(x)}}&=c_1,y(x)\right ] \\ \text {Solve}\left [\frac {\sqrt {x^6+4 x^2 y(x)} \text {arctanh}\left (\frac {\sqrt {x^4+4 y(x)}}{x^2}\right )}{2 x \sqrt {x^4+4 y(x)}}+\frac {1}{4} \log (y(x))&=c_1,y(x)\right ] \\ y(x)\to -\frac {x^4}{4} \\ \end{align*}

✓ Sympy. Time used: 1.600 (sec). Leaf size: 8

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(2*x**3*Derivative(y(x), x) - 4*x**2*y(x) + Derivative(y(x), x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = C_{1} \left (C_{1} + x^{2}\right ) \]