75.8.18 problem 216

Internal problem ID [16763]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Section 8. First order not solved for the derivative. Exercises page 67
Problem number : 216
Date solved : Monday, March 31, 2025 at 03:16:22 PM
CAS classification : [_quadrature]

\begin{align*} y^{{2}/{5}}+{y^{\prime }}^{{2}/{5}}&=a^{{2}/{5}} \end{align*}

✓ Maple. Time used: 0.103 (sec). Leaf size: 26

ode:=y(x)^(2/5)+diff(y(x),x)^(2/5) = a^(2/5); 
dsolve(ode,y(x), singsol=all);
 

\[ x -\int _{}^{y}\frac {1}{\left (a^{{2}/{5}}-\textit {\_a}^{{2}/{5}}\right )^{{5}/{2}}}d \textit {\_a} -c_1 = 0 \]

✓ Mathematica. Time used: 0.876 (sec). Leaf size: 113

ode=y[x]^(2/5)+D[y[x],x]^(2/5)==a^(2/5); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to \text {InverseFunction}\left [\frac {15 \sqrt [5]{a} \sqrt {1-\frac {\text {$\#$1}^{2/5}}{a^{2/5}}} \left (a^{2/5}-\text {$\#$1}^{2/5}\right ) \arcsin \left (\frac {\sqrt [5]{\text {$\#$1}}}{\sqrt [5]{a}}\right )+20 \text {$\#$1}^{3/5}-15 \sqrt [5]{\text {$\#$1}} a^{2/5}}{3 \left (a^{2/5}-\text {$\#$1}^{2/5}\right )^{3/2}}\&\right ][x+c_1] \\ y(x)\to a \\ \end{align*}

✓ Sympy. Time used: 11.032 (sec). Leaf size: 112

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(-a**(2/5) + y(x)**(2/5) + Derivative(y(x), x)**(2/5),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ \int \limits ^{y{\left (x \right )}} \frac {1}{\sqrt {- \left (\sqrt [5]{y} - \sqrt [5]{a}\right ) \left (\sqrt [5]{y} + \sqrt [5]{a}\right )} \left (\sqrt [5]{y} - \sqrt [5]{a}\right )^{2} \left (\sqrt [5]{y} + \sqrt [5]{a}\right )^{2}}\, dy = C_{1} - x, \ \int \limits ^{y{\left (x \right )}} \frac {1}{\sqrt {- \left (\sqrt [5]{y} - \sqrt [5]{a}\right ) \left (\sqrt [5]{y} + \sqrt [5]{a}\right )} \left (\sqrt [5]{y} - \sqrt [5]{a}\right )^{2} \left (\sqrt [5]{y} + \sqrt [5]{a}\right )^{2}}\, dy = C_{1} + x\right ] \]