75.11.5 problem 264

Internal problem ID [16856]
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 : Tuesday, January 28, 2025 at 09:35:17 AM
CAS classification : [_quadrature]

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

✓ Solution by Maple

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

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

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

✓ Solution by Mathematica

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

DSolve[D[y[x],x]==y[x]^(2/3)+a,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*}