40.6.6 problem 15

Internal problem ID [6686]
Book : Schaums Outline. Theory and problems of Differential Equations, 1st edition. Frank Ayres. McGraw Hill 1952
Section : Chapter 10. Singular solutions, Extraneous loci. Supplemetary problems. Page 74
Problem number : 15
Date solved : Wednesday, March 05, 2025 at 02:35:25 AM
CAS classification : [[_homogeneous, `class G`], _rational]

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

✓ Maple. Time used: 0.024 (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 {-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: 0.535 (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: 4.604 (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)
 

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