64.3.15 problem 16

Internal problem ID [13215]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 2, section 2.1 (Exact differential equations and integrating factors). Exercises page 37
Problem number : 16
Date solved : Monday, March 31, 2025 at 07:39:33 AM
CAS classification : [[_homogeneous, `class G`], _exact, _rational]

\begin{align*} \frac {1+8 x y^{{2}/{3}}}{x^{{2}/{3}} y^{{1}/{3}}}+\frac {\left (2 x^{{4}/{3}} y^{{2}/{3}}-x^{{1}/{3}}\right ) y^{\prime }}{y^{{4}/{3}}}&=0 \end{align*}

With initial conditions

\begin{align*} y \left (1\right )&=8 \end{align*}

✓ Maple. Time used: 0.216 (sec). Leaf size: 39

ode:=(1+8*x*y(x)^(2/3))/x^(2/3)/y(x)^(1/3)+(2*x^(4/3)*y(x)^(2/3)-x^(1/3))/y(x)^(4/3)*diff(y(x),x) = 0; 
ic:=y(1) = 8; 
dsolve([ode,ic],y(x), singsol=all);
 

\[ y = \operatorname {RootOf}\left (64 \textit {\_Z}^{{7}/{3}} x^{4}+96 \textit {\_Z}^{{5}/{3}} x^{3}-729 \textit {\_Z}^{{4}/{3}}+48 \textit {\_Z} \,x^{2}+8 x \,\textit {\_Z}^{{1}/{3}}\right ) \]

✓ Mathematica. Time used: 0.614 (sec). Leaf size: 207

ode=(1+8*x*y[x]^(2/3))/(x^(2/3)*y[x]^(1/3))+((2*x^(4/3)*y[x]^(2/3)-x^(1/3))/(y[x]^(4/3)))*D[y[x],x]==0; 
ic={y[1]==8}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [\int _1^{y(x)}\left (-\frac {16 \sqrt [3]{K[2]} x^2}{5 \left (2 K[2]^{2/3} x+1\right )}+\frac {8 x}{5 \sqrt [3]{K[2]}}-\int _1^x\left (\frac {8}{5 \left (2 K[2]^{2/3} K[1]+1\right ) \sqrt [3]{K[2]}}-\frac {16 K[1] \sqrt [3]{K[2]}}{5 \left (2 K[2]^{2/3} K[1]+1\right )^2}\right )dK[1]-\frac {2}{5 K[2]}\right )dK[2]+\int _1^x\left (\frac {12 y(x)^{2/3}}{5 \left (2 y(x)^{2/3} K[1]+1\right )}+\frac {2}{5 K[1]}\right )dK[1]=\int _1^8\left (-\frac {16 \sqrt [3]{K[2]}}{5 \left (2 K[2]^{2/3}+1\right )}+\frac {8}{5 \sqrt [3]{K[2]}}-\frac {2}{5 K[2]}\right )dK[2],y(x)\right ] \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((2*x**(4/3)*y(x)**(2/3) - x**(1/3))*Derivative(y(x), x)/y(x)**(4/3) + (8*x*y(x)**(2/3) + 1)/(x**(2/3)*y(x)**(1/3)),0) 
ics = {y(1): 8} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(x), x) - (-8*x*y(x)**(5/3) - y(x))/(x*(2*x*y(x)**(2/3) - 1)) cannot be solved by the factorable group method