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