62.14.5 problem Ex 5

Internal problem ID [12807]
Book : An elementary treatise on differential equations by Abraham Cohen. DC heath publishers. 1906
Section : Chapter IV, differential equations of the first order and higher degree than the first. Article 25. Equations solvable for \(y\). Page 52
Problem number : Ex 5
Date solved : Wednesday, March 05, 2025 at 08:32:55 PM
CAS classification : [[_homogeneous, `class G`], _rational]

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

✓ Maple. Time used: 3.109 (sec). Leaf size: 77

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

✓ Mathematica. Time used: 0.524 (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: 4.193 (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)
 

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