75.11.5 problem 264

Internal problem ID [16785]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Section 11. Singular solutions of differential equations. Exercises page 92
Problem number : 264
Date solved : Monday, March 31, 2025 at 03:17:33 PM
CAS classification : [_quadrature]

\begin{align*} y^{\prime }&=y^{{2}/{3}}+a \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 85

ode:=diff(y(x),x) = y(x)^(2/3)+a; 
dsolve(ode,y(x), singsol=all);
 

\[ x -3 y^{{1}/{3}}+\sqrt {a}\, \arctan \left (\frac {2 y^{{1}/{3}}+\sqrt {3}\, \sqrt {a}}{\sqrt {a}}\right )-\sqrt {a}\, \arctan \left (\frac {\sqrt {3}\, \sqrt {a}-2 y^{{1}/{3}}}{\sqrt {a}}\right )+2 \sqrt {a}\, \arctan \left (\frac {y^{{1}/{3}}}{\sqrt {a}}\right )-\sqrt {a}\, \arctan \left (\frac {y}{a^{{3}/{2}}}\right )+c_1 = 0 \]

✓ Mathematica. Time used: 0.321 (sec). Leaf size: 51

ode=D[y[x],x]==y[x]^(2/3)+a; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 1.255 (sec). Leaf size: 75

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(-a - y(x)**(2/3) + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \begin {cases} \frac {3 a \log {\left (- \sqrt {- a} + \sqrt [3]{y{\left (x \right )}} \right )}}{2 \sqrt {- a}} - \frac {3 a \log {\left (\sqrt {- a} + \sqrt [3]{y{\left (x \right )}} \right )}}{2 \sqrt {- a}} - 3 \sqrt [3]{y{\left (x \right )}} & \text {for}\: a \neq 0 \\- 3 \sqrt [3]{y{\left (x \right )}} & \text {otherwise} \end {cases} = C_{1} - x \]