2.1.53 Problem 53

Internal problem ID [9037]
Book : First order enumerated odes
Section : section 1
Problem number : 53
Date solved : Friday, February 21, 2025 at 09:03:49 PM
CAS classification : [[_homogeneous, `class G`]]

Solve

\begin{align*} {y^{\prime }}^{2}&=\frac {y^{3}}{x} \end{align*}

Solved as first order ode of type nonlinear p but separable

Time used: 0.376 (sec)

The ode has the form

\begin{align*} (y')^{\frac {n}{m}} &= f(x) g(y)\tag {1} \end{align*}

Where \(n=2, m=1, f=\frac {1}{x} , g=y^{3}\). Hence the ode is

\begin{align*} (y')^{2} &= \frac {y^{3}}{x} \end{align*}

Solving for \(y^{\prime }\) from (1) gives

\begin{align*} y^{\prime } &=\sqrt {f g}\\ y^{\prime } &=-\sqrt {f g} \end{align*}

To be able to solve as separable ode, we have to now assume that \(f>0,g>0\).

\begin{align*} \frac {1}{x} &> 0\\ y^{3} &> 0 \end{align*}

Under the above assumption the differential equations become separable and can be written as

\begin{align*} y^{\prime } &=\sqrt {f}\, \sqrt {g}\\ y^{\prime } &=-\sqrt {f}\, \sqrt {g} \end{align*}

Therefore

\begin{align*} \frac {1}{\sqrt {g}} \, dy &= \left (\sqrt {f}\right )\,dx\\ -\frac {1}{\sqrt {g}} \, dy &= \left (\sqrt {f}\right )\,dx \end{align*}

Replacing \(f(x),g(y)\) by their values gives

\begin{align*} \frac {1}{\sqrt {y^{3}}} \, dy &= \left (\sqrt {\frac {1}{x}}\right )\,dx\\ -\frac {1}{\sqrt {y^{3}}} \, dy &= \left (\sqrt {\frac {1}{x}}\right )\,dx \end{align*}

Integrating now gives the following solutions

\begin{align*} \int \frac {1}{\sqrt {y^{3}}}d y &= \int \sqrt {\frac {1}{x}}d x +c_1\\ -\frac {2 \sqrt {y^{3}}}{y^{2}} &= 2 x \sqrt {\frac {1}{x}}\\ \int -\frac {1}{\sqrt {y^{3}}}d y &= \int \sqrt {\frac {1}{x}}d x +c_1\\ \frac {2 \sqrt {y^{3}}}{y^{2}} &= 2 x \sqrt {\frac {1}{x}} \end{align*}

Therefore

\begin{align*} y &= \frac {4}{4 x \sqrt {\frac {1}{x}}\, c_1 +c_1^{2}+4 x} \\ y &= \frac {4}{4 x \sqrt {\frac {1}{x}}\, c_1 +c_1^{2}+4 x} \\ \end{align*}

Summary of solutions found

\begin{align*} y &= \frac {4}{4 x \sqrt {\frac {1}{x}}\, c_1 +c_1^{2}+4 x} \\ \end{align*}

Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left (\frac {d}{d x}y \left (x \right )\right )^{2}=\frac {y \left (x \right )^{3}}{x} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \left [\frac {d}{d x}y \left (x \right )=\frac {\sqrt {x y \left (x \right )}\, y \left (x \right )}{x}, \frac {d}{d x}y \left (x \right )=-\frac {\sqrt {x y \left (x \right )}\, y \left (x \right )}{x}\right ] \\ \bullet & {} & \textrm {Solve the equation}\hspace {3pt} \frac {d}{d x}y \left (x \right )=\frac {\sqrt {x y \left (x \right )}\, y \left (x \right )}{x} \\ \bullet & {} & \textrm {Solve the equation}\hspace {3pt} \frac {d}{d x}y \left (x \right )=-\frac {\sqrt {x y \left (x \right )}\, y \left (x \right )}{x} \\ \bullet & {} & \textrm {Set of solutions}\hspace {3pt} \\ {} & {} & \left \{\mathit {workingODE} , \mathit {workingODE}\right \} \end {array} \]

Maple trace

`Methods for first order ODEs: 
-> Solving 1st order ODE of high degree, 1st attempt 
trying 1st order WeierstrassP solution for high degree ODE 
<- 1st_order WeierstrassP successful`
 

Maple dsolve solution

Solving time : 0.030 (sec)
Leaf size : 27

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

\begin{align*} y &= 0 \\ y &= \frac {\operatorname {WeierstrassP}\left (1, 0, 0\right ) 2^{{2}/{3}}}{\left (\sqrt {x}\, 2^{{1}/{3}}+c_{1} \right )^{2}} \\ \end{align*}

✓Mathematica DSolve solution

Solving time : 0.076 (sec)
Leaf size : 42

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

\begin{align*} y(x)\to \frac {4}{\left (-2 \sqrt {x}+c_1\right ){}^2} \\ y(x)\to \frac {4}{\left (2 \sqrt {x}+c_1\right ){}^2} \\ y(x)\to 0 \\ \end{align*}

✓Sympy solution

Solving time : 11.699 (sec)
Leaf size : 80

Python version: 3.13.1 (main, Dec  4 2024, 18:05:56) [GCC 14.2.1 20240910] 
Sympy version 1.13.3
 

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

Eq(y(x), (C1**6 - 2*C1**5*sqrt(x) - C1**4*x + 4*C1**3*x**(3/2) - 
C1**2*x**2 - 2*C1*x**(5/2) + x**3)/(C1**8 - 4*C1**6*x + 6*C1**4*x**2 - 
4*C1**2*x**3 + x**4))
 

\[ y{\left (x \right )} = \frac {C_{1}^{6} - 2 C_{1}^{5} \sqrt {x} - C_{1}^{4} x + 4 C_{1}^{3} x^{\frac {3}{2}} - C_{1}^{2} x^{2} - 2 C_{1} x^{\frac {5}{2}} + x^{3}}{C_{1}^{8} - 4 C_{1}^{6} x + 6 C_{1}^{4} x^{2} - 4 C_{1}^{2} x^{3} + x^{4}} \]