23.2.204 problem 209

Internal problem ID [5559]
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 : 209
Date solved : Tuesday, September 30, 2025 at 12:53:15 PM
CAS classification : [[_homogeneous, `class C`], _rational, _dAlembert]

\begin{align*} 2 y {y^{\prime }}^{2}+\left (5-4 x \right ) y^{\prime }+2 y&=0 \end{align*}

✓ Maple. Time used: 0.154 (sec). Leaf size: 119

ode:=2*y(x)*diff(y(x),x)^2+(5-4*x)*diff(y(x),x)+2*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= x -\frac {5}{4} \\ y &= -x +\frac {5}{4} \\ y &= 0 \\ y &= \frac {\sqrt {4 c_1 +2 \sqrt {-c_1 \left (-5+4 x \right )^{2}}}}{2} \\ y &= -\frac {\sqrt {4 c_1 +2 \sqrt {-c_1 \left (-5+4 x \right )^{2}}}}{2} \\ y &= \frac {\sqrt {4 c_1 -2 \sqrt {-c_1 \left (-5+4 x \right )^{2}}}}{2} \\ y &= -\frac {\sqrt {4 c_1 -2 \sqrt {-c_1 \left (-5+4 x \right )^{2}}}}{2} \\ \end{align*}

✓ Mathematica. Time used: 0.386 (sec). Leaf size: 160

ode=(2 y[x] (D[y[x],x])^2)+(5-4 x)D[y[x],x]+2 y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -i \sqrt {2} e^{\frac {c_1}{2}} \sqrt {4 x-5+8 e^{c_1}}\\ y(x)&\to i \sqrt {2} e^{\frac {c_1}{2}} \sqrt {4 x-5+8 e^{c_1}}\\ y(x)&\to -\frac {1}{4} i e^{\frac {c_1}{2}} \sqrt {8 x-10+e^{c_1}}\\ y(x)&\to \frac {1}{4} i e^{\frac {c_1}{2}} \sqrt {8 x-10+e^{c_1}}\\ y(x)&\to 0\\ y(x)&\to \frac {5}{4}-x\\ y(x)&\to x-\frac {5}{4} \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((5 - 4*x)*Derivative(y(x), x) + 2*y(x)*Derivative(y(x), x)**2 + 2*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out