2.1.55 problem 55
Internal
problem
ID
[8715]
Book
:
First
order
enumerated
odes
Section
:
section
1
Problem
number
:
55
Date
solved
:
Tuesday, December 17, 2024 at 12:58:21 PM
CAS
classification
:
[[_homogeneous, `class G`]]
Solve
\begin{align*} {y^{\prime }}^{2}&=\frac {1}{y x} \end{align*}
Solved as first order ode of type nonlinear p but separable
Time used: 0.305 (sec)
The ode has the form
\begin{align*} (y')^{\frac {n}{m}} &= f(x) g(y)\tag {1} \end{align*}
Where \(n=2, m=1, f=\frac {1}{x} , g=\frac {1}{y}\). Hence the ode is
\begin{align*} (y')^{2} &= \frac {1}{y 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\\ \frac {1}{y} &> 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 {\frac {1}{y}}} \, dy &= \left (\sqrt {\frac {1}{x}}\right )\,dx\\ -\frac {1}{\sqrt {\frac {1}{y}}} \, dy &= \left (\sqrt {\frac {1}{x}}\right )\,dx \end{align*}
Integrating now gives the following solutions
\begin{align*} \int \frac {1}{\sqrt {\frac {1}{y}}}d y &= \int \sqrt {\frac {1}{x}}d x +c_1\\ \frac {2 y^{2} \sqrt {\frac {1}{y}}}{3} &= 2 x \sqrt {\frac {1}{x}}\\ \int -\frac {1}{\sqrt {\frac {1}{y}}}d y &= \int \sqrt {\frac {1}{x}}d x +c_1\\ -\frac {2 y^{2} \sqrt {\frac {1}{y}}}{3} &= 2 x \sqrt {\frac {1}{x}} \end{align*}
Therefore
\begin{align*}
\frac {2 y^{2} \sqrt {\frac {1}{y}}}{3} &= 2 x \sqrt {\frac {1}{x}}+c_1 \\
-\frac {2 y^{2} \sqrt {\frac {1}{y}}}{3} &= 2 x \sqrt {\frac {1}{x}}+c_1 \\
\end{align*}
Solving for \(y\) gives
\begin{align*}
y &= \frac {1}{{\left (-\frac {\left (-18 \left (2 x \sqrt {\frac {1}{x}}+c_1 \right )^{2}\right )^{{1}/{3}}}{6 \left (2 x \sqrt {\frac {1}{x}}+c_1 \right )}-\frac {i \sqrt {3}\, \left (-18 \left (2 x \sqrt {\frac {1}{x}}+c_1 \right )^{2}\right )^{{1}/{3}}}{6 \left (2 x \sqrt {\frac {1}{x}}+c_1 \right )}\right )}^{2}} \\
y &= \frac {1}{{\left (-\frac {\left (-18 \left (2 x \sqrt {\frac {1}{x}}+c_1 \right )^{2}\right )^{{1}/{3}}}{6 \left (2 x \sqrt {\frac {1}{x}}+c_1 \right )}+\frac {i \sqrt {3}\, \left (-18 \left (2 x \sqrt {\frac {1}{x}}+c_1 \right )^{2}\right )^{{1}/{3}}}{12 x \sqrt {\frac {1}{x}}+6 c_1}\right )}^{2}} \\
y &= \frac {1}{\left (-\frac {18^{{1}/{3}} \left (\left (2 x \sqrt {\frac {1}{x}}+c_1 \right )^{2}\right )^{{1}/{3}}}{6 \left (2 x \sqrt {\frac {1}{x}}+c_1 \right )}-\frac {i \sqrt {3}\, 18^{{1}/{3}} \left (\left (2 x \sqrt {\frac {1}{x}}+c_1 \right )^{2}\right )^{{1}/{3}}}{6 \left (2 x \sqrt {\frac {1}{x}}+c_1 \right )}\right )^{2}} \\
y &= \frac {1}{\left (-\frac {18^{{1}/{3}} \left (\left (2 x \sqrt {\frac {1}{x}}+c_1 \right )^{2}\right )^{{1}/{3}}}{6 \left (2 x \sqrt {\frac {1}{x}}+c_1 \right )}+\frac {i \sqrt {3}\, 18^{{1}/{3}} \left (\left (2 x \sqrt {\frac {1}{x}}+c_1 \right )^{2}\right )^{{1}/{3}}}{12 x \sqrt {\frac {1}{x}}+6 c_1}\right )^{2}} \\
y &= \frac {9 \left (2 x \sqrt {\frac {1}{x}}+c_1 \right )^{2}}{\left (-18 \left (2 x \sqrt {\frac {1}{x}}+c_1 \right )^{2}\right )^{{2}/{3}}} \\
y &= \frac {\left (2 x \sqrt {\frac {1}{x}}+c_1 \right )^{2} 18^{{1}/{3}}}{2 \left (\left (2 x \sqrt {\frac {1}{x}}+c_1 \right )^{2}\right )^{{2}/{3}}} \\
\end{align*}
Summary of solutions found
\begin{align*}
y &= \frac {1}{{\left (-\frac {\left (-18 \left (2 x \sqrt {\frac {1}{x}}+c_1 \right )^{2}\right )^{{1}/{3}}}{6 \left (2 x \sqrt {\frac {1}{x}}+c_1 \right )}-\frac {i \sqrt {3}\, \left (-18 \left (2 x \sqrt {\frac {1}{x}}+c_1 \right )^{2}\right )^{{1}/{3}}}{6 \left (2 x \sqrt {\frac {1}{x}}+c_1 \right )}\right )}^{2}} \\
y &= \frac {1}{{\left (-\frac {\left (-18 \left (2 x \sqrt {\frac {1}{x}}+c_1 \right )^{2}\right )^{{1}/{3}}}{6 \left (2 x \sqrt {\frac {1}{x}}+c_1 \right )}+\frac {i \sqrt {3}\, \left (-18 \left (2 x \sqrt {\frac {1}{x}}+c_1 \right )^{2}\right )^{{1}/{3}}}{12 x \sqrt {\frac {1}{x}}+6 c_1}\right )}^{2}} \\
y &= \frac {1}{\left (-\frac {18^{{1}/{3}} \left (\left (2 x \sqrt {\frac {1}{x}}+c_1 \right )^{2}\right )^{{1}/{3}}}{6 \left (2 x \sqrt {\frac {1}{x}}+c_1 \right )}-\frac {i \sqrt {3}\, 18^{{1}/{3}} \left (\left (2 x \sqrt {\frac {1}{x}}+c_1 \right )^{2}\right )^{{1}/{3}}}{6 \left (2 x \sqrt {\frac {1}{x}}+c_1 \right )}\right )^{2}} \\
y &= \frac {1}{\left (-\frac {18^{{1}/{3}} \left (\left (2 x \sqrt {\frac {1}{x}}+c_1 \right )^{2}\right )^{{1}/{3}}}{6 \left (2 x \sqrt {\frac {1}{x}}+c_1 \right )}+\frac {i \sqrt {3}\, 18^{{1}/{3}} \left (\left (2 x \sqrt {\frac {1}{x}}+c_1 \right )^{2}\right )^{{1}/{3}}}{12 x \sqrt {\frac {1}{x}}+6 c_1}\right )^{2}} \\
y &= \frac {9 \left (2 x \sqrt {\frac {1}{x}}+c_1 \right )^{2}}{\left (-18 \left (2 x \sqrt {\frac {1}{x}}+c_1 \right )^{2}\right )^{{2}/{3}}} \\
y &= \frac {\left (2 x \sqrt {\frac {1}{x}}+c_1 \right )^{2} 18^{{1}/{3}}}{2 \left (\left (2 x \sqrt {\frac {1}{x}}+c_1 \right )^{2}\right )^{{2}/{3}}} \\
\end{align*}
Maple step by step solution
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left (\frac {d}{d x}y \left (x \right )\right )^{2}=\frac {1}{x y \left (x \right )} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \left [\frac {d}{d x}y \left (x \right )=\frac {1}{\sqrt {x y \left (x \right )}}, \frac {d}{d x}y \left (x \right )=-\frac {1}{\sqrt {x y \left (x \right )}}\right ] \\ \bullet & {} & \textrm {Solve the equation}\hspace {3pt} \frac {d}{d x}y \left (x \right )=\frac {1}{\sqrt {x y \left (x \right )}} \\ \bullet & {} & \textrm {Solve the equation}\hspace {3pt} \frac {d}{d x}y \left (x \right )=-\frac {1}{\sqrt {x y \left (x \right )}} \\ \bullet & {} & \textrm {Set of solutions}\hspace {3pt} \\ {} & {} & \left \{\mathit {workingODE} , \mathit {workingODE}\right \} \end {array} \]
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
trying 1st order WeierstrassPPrime solution for high degree ODE
trying 1st order JacobiSN solution for high degree ODE
trying 1st order ODE linearizable_by_differentiation
trying differential order: 1; missing variables
trying simple symmetries for implicit equations
Successful isolation of dy/dx: 2 solutions were found. Trying to solve each resulting ODE.
*** Sublevel 2 ***
Methods for first order ODEs:
--- Trying classification methods ---
trying homogeneous types:
trying homogeneous G
1st order, trying the canonical coordinates of the invariance group
<- 1st order, canonical coordinates successful
<- homogeneous successful
-------------------
* Tackling next ODE.
*** Sublevel 2 ***
Methods for first order ODEs:
--- Trying classification methods ---
trying homogeneous types:
trying homogeneous G
1st order, trying the canonical coordinates of the invariance group
<- 1st order, canonical coordinates successful
<- homogeneous successful`
Maple dsolve solution
Solving time : 0.085 (sec)
Leaf size : 51
dsolve(diff(y(x),x)^2 = 1/x/y(x),
y(x),singsol=all)
\begin{align*}
\frac {y \sqrt {x y}-c_{1} \sqrt {x}-3 x}{\sqrt {x}} &= 0 \\
\frac {y \sqrt {x y}-c_{1} \sqrt {x}+3 x}{\sqrt {x}} &= 0 \\
\end{align*}
Mathematica DSolve solution
Solving time : 3.342 (sec)
Leaf size : 53
DSolve[{(D[y[x],x])^2==1/(y[x]*x),{}},
y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to \left (\frac {3}{2}\right )^{2/3} \left (-2 \sqrt {x}+c_1\right ){}^{2/3} \\
y(x)\to \left (\frac {3}{2}\right )^{2/3} \left (2 \sqrt {x}+c_1\right ){}^{2/3} \\
\end{align*}