83.23.24 problem 24

Internal problem ID [19226]
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 : 24
Date solved : Thursday, March 13, 2025 at 01:59:35 PM
CAS classification : [_quadrature]

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

✓ Maple. Time used: 0.046 (sec). Leaf size: 137

ode:=diff(y(x),x)^2 = (4*y(x)+1)*(diff(y(x),x)-y(x)); 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.193 (sec). Leaf size: 67

ode=D[y[x],x]^2==(4*y[x]+1)*(D[y[x],x]-y[x]); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to \frac {1}{4} e^{x-4 c_1} \left (e^x+2 e^{2 c_1}\right ) \\ y(x)\to \frac {1}{4} e^{x+2 c_1} \left (-2+e^{x+2 c_1}\right ) \\ y(x)\to -\frac {1}{4} \\ y(x)\to 0 \\ \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((y(x) - Derivative(y(x), x))*(4*y(x) + 1) + Derivative(y(x), x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out