44.1.11 problem 1(L)

Internal problem ID [9069]
Book : Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section : Chapter 1. What is a differential equation. Section 1.2 THE NATURE OF SOLUTIONS. Page 9
Problem number : 1(L)
Date solved : Tuesday, September 30, 2025 at 06:03:36 PM
CAS classification : [[_homogeneous, `class G`], _rational]

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

✓ Maple. Time used: 0.177 (sec). Leaf size: 81

ode:=x*diff(y(x),x)+y(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: 0.317 (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: 2.453 (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)
 

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