15.20.9 problem 9

Internal problem ID [3317]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 38, page 173
Problem number : 9
Date solved : Sunday, March 30, 2025 at 01:34:50 AM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

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

✓ Maple. Time used: 0.031 (sec). Leaf size: 35

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

\begin{align*} y &= -\frac {x}{4} \\ y &= 4 c_1 +2 \sqrt {c_1 x} \\ y &= 4 c_1 -2 \sqrt {c_1 x} \\ \end{align*}

✓ Mathematica. Time used: 0.156 (sec). Leaf size: 72

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

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

✓ Sympy. Time used: 31.646 (sec). Leaf size: 92

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

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