1.55 problem 55

Internal problem ID [8019]
Book : First order enumerated odes
Section : section 1
Problem number : 55
Date solved : Monday, October 21, 2024 at 04:41:04 PM
CAS classification : [[_homogeneous, `class G`]]

Solve

\begin{align*} {y^{\prime }}^{2}&=\frac {1}{y x} \end{align*}

1.55.1 Solved as first order ode of type nonlinear p but separable

Time used: 0.299 (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 \(y\) from the above solution(s) gives (after possible removing of solutions that do not verify)

\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*}

1.55.2 Maple step by step solution

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

1.55.3 Maple trace

Methods for first order ODEs:
 

1.55.4 Maple dsolve solution

Solving time : 0.037 (sec)
Leaf size : 51

dsolve(diff(y(x),x)^2 = 1/y(x)/x, 
       y(x),singsol=all)
 

\begin{align*} \frac {y \sqrt {y x}-c_1 \sqrt {x}-3 x}{\sqrt {x}} &= 0 \\ \frac {y \sqrt {y x}-c_1 \sqrt {x}+3 x}{\sqrt {x}} &= 0 \\ \end{align*}

1.55.5 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*}