23.2.65 problem 67

Internal problem ID [5420]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 2. THE DIFFERENTIAL EQUATION IS OF FIRST ORDER AND OF SECOND OR HIGHER DEGREE, page 278
Problem number : 67
Date solved : Tuesday, September 30, 2025 at 12:41:00 PM
CAS classification : [_quadrature]

\begin{align*} {y^{\prime }}^{2}-\left (1+4 y\right ) y^{\prime }+\left (1+4 y\right ) y&=0 \end{align*}

✓ Maple. Time used: 0.126 (sec). Leaf size: 137

ode:=diff(y(x),x)^2-(1+4*y(x))*diff(y(x),x)+(1+4*y(x))*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= -{\frac {1}{4}} \\ y &= -\frac {\sqrt {-{\mathrm e}^{-2 x} c_1}\, {\mathrm e}^{2 x}+c_1}{c_1 \sqrt {-{\mathrm e}^{-2 x} c_1}} \\ y &= \frac {-\sqrt {-{\mathrm e}^{-2 x} c_1}\, {\mathrm e}^{2 x}+c_1}{\sqrt {-{\mathrm e}^{-2 x} c_1}\, c_1} \\ y &= \frac {-\sqrt {-{\mathrm e}^{-2 x} c_1}\, {\mathrm e}^{2 x}+c_1}{\sqrt {-{\mathrm e}^{-2 x} c_1}\, c_1} \\ y &= -\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: 0.114 (sec). Leaf size: 67

ode=(D[y[x],x])^2-(1+4*y[x])*D[y[x],x]+(1+4*y[x])*y[x]==0; 
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((4*y(x) + 1)*y(x) - (4*y(x) + 1)*Derivative(y(x), x) + Derivative(y(x), x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out