76.18.4 problem Ex. 4

Internal problem ID [20111]
Book : Introductory Course On Differential Equations by Daniel A Murray. Longmans Green and Co. NY. 1924
Section : Chapter III. Equations of the first order but not of the first degree. Examples on chapter III. page 38
Problem number : Ex. 4
Date solved : Thursday, October 02, 2025 at 05:26:00 PM
CAS classification : [[_homogeneous, `class G`], _rational]

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

✓ Maple. Time used: 0.180 (sec). Leaf size: 81

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

\begin{align*} y &= -\frac {1}{4 x^{2}} \\ y &= \frac {i c_1 -x}{x \,c_1^{2}} \\ y &= \frac {i c_1 -x}{x \,c_1^{2}} \\ y &= \frac {-i c_1 -x}{x \,c_1^{2}} \\ y &= \frac {-i c_1 -x}{x \,c_1^{2}} \\ \end{align*}

✓ Mathematica. Time used: 0.316 (sec). Leaf size: 123

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

\begin{align*} \text {Solve}\left [-\frac {x \sqrt {4 x^2 y(x)+1} \text {arctanh}\left (\sqrt {4 x^2 y(x)+1}\right )}{\sqrt {4 x^4 y(x)+x^2}}-\frac {1}{2} \log (y(x))=c_1,y(x)\right ]\\ \text {Solve}\left [\frac {x \sqrt {4 x^2 y(x)+1} \text {arctanh}\left (\sqrt {4 x^2 y(x)+1}\right )}{\sqrt {4 x^4 y(x)+x^2}}-\frac {1}{2} \log (y(x))=c_1,y(x)\right ]\\ y(x)&\to 0 \end{align*}

✓ Sympy. Time used: 2.495 (sec). Leaf size: 10

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

\[ y{\left (x \right )} = \frac {C_{1} \left (C_{1} + \frac {2}{x}\right )}{4} \]