83.22.8 problem 8

Internal problem ID [19191]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Chapter IV. Equations of the first order but not of the first degree. Exercise IV (E) at page 63
Problem number : 8
Date solved : Monday, March 31, 2025 at 06:53:14 PM
CAS classification : [[_homogeneous, `class G`]]

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

✓ Maple. Time used: 0.192 (sec). Leaf size: 86

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

\begin{align*} y &= \frac {1}{\sqrt {-x}} \\ y &= -\frac {1}{\sqrt {-x}} \\ y &= 0 \\ y &= \frac {\sqrt {x \operatorname {sech}\left (-\frac {\ln \left (x \right )}{2}+\frac {c_1}{2}\right )^{2}}\, \coth \left (-\frac {\ln \left (x \right )}{2}+\frac {c_1}{2}\right )}{x} \\ y &= -\frac {\sqrt {x \operatorname {sech}\left (-\frac {\ln \left (x \right )}{2}+\frac {c_1}{2}\right )^{2}}\, \coth \left (-\frac {\ln \left (x \right )}{2}+\frac {c_1}{2}\right )}{x} \\ \end{align*}

✓ Mathematica. Time used: 0.493 (sec). Leaf size: 80

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

\begin{align*} y(x)\to -\frac {2 e^{\frac {c_1}{2}}}{-x+e^{c_1}} \\ y(x)\to \frac {2 e^{\frac {c_1}{2}}}{-x+e^{c_1}} \\ y(x)\to 0 \\ y(x)\to -\frac {i}{\sqrt {x}} \\ y(x)\to \frac {i}{\sqrt {x}} \\ \end{align*}

✓ Sympy. Time used: 18.474 (sec). Leaf size: 36

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

\[ \left [ y{\left (x \right )} = \frac {1}{\sqrt {x} \sinh {\left (C_{1} + \frac {\log {\left (x \right )}}{2} \right )}}, \ y{\left (x \right )} = - \frac {1}{\sqrt {x} \sinh {\left (C_{1} + \frac {\log {\left (x \right )}}{2} \right )}}\right ] \]