1.57 problem 57

Internal problem ID [8021]
Book : First order enumerated odes
Section : section 1
Problem number : 57
Date solved : Monday, October 21, 2024 at 04:41:07 PM
CAS classification : [_separable]

Solve

\begin{align*} {y^{\prime }}^{2}&=\frac {1}{x^{2} y^{3}} \end{align*}

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

Time used: 0.331 (sec)

The ode has the form

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

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

\begin{align*} (y')^{2} &= \frac {1}{x^{2} y^{3}} \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^{2}} &> 0\\ \frac {1}{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 {\frac {1}{y^{3}}}} \, dy &= \left (\sqrt {\frac {1}{x^{2}}}\right )\,dx\\ -\frac {1}{\sqrt {\frac {1}{y^{3}}}} \, dy &= \left (\sqrt {\frac {1}{x^{2}}}\right )\,dx \end{align*}

Integrating now gives the following solutions

\begin{align*} \int \frac {1}{\sqrt {\frac {1}{y^{3}}}}d y &= \int \sqrt {\frac {1}{x^{2}}}d x +c_1\\ \frac {2 y^{4} \sqrt {\frac {1}{y^{3}}}}{5} &= \sqrt {\frac {1}{x^{2}}}\, x \ln \left (x \right )\\ \int -\frac {1}{\sqrt {\frac {1}{y^{3}}}}d y &= \int \sqrt {\frac {1}{x^{2}}}d x +c_1\\ -\frac {2 y^{4} \sqrt {\frac {1}{y^{3}}}}{5} &= \sqrt {\frac {1}{x^{2}}}\, x \ln \left (x \right ) \end{align*}

Therefore

\begin{align*} \frac {2 y^{4} \sqrt {\frac {1}{y^{3}}}}{5} &= \sqrt {\frac {1}{x^{2}}}\, x \ln \left (x \right )+c_1 \\ -\frac {2 y^{4} \sqrt {\frac {1}{y^{3}}}}{5} &= \sqrt {\frac {1}{x^{2}}}\, x \ln \left (x \right )+c_1 \\ \end{align*}

1.57.2 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & {y^{\prime }}^{2}=\frac {1}{x^{2} y^{3}} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \left [y^{\prime }=\frac {1}{y^{{3}/{2}} x}, y^{\prime }=-\frac {1}{y^{{3}/{2}} x}\right ] \\ \square & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=\frac {1}{y^{{3}/{2}} x} \\ {} & \circ & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & y^{\prime } y^{{3}/{2}}=\frac {1}{x} \\ {} & \circ & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int y^{\prime } y^{{3}/{2}}d x =\int \frac {1}{x}d x +\textit {\_C1} \\ {} & \circ & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \frac {2 y^{{5}/{2}}}{5}=\ln \left (x \right )+\textit {\_C1} \\ {} & \circ & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\frac {\left (80 \ln \left (x \right )+80 \textit {\_C1} \right )^{{2}/{5}}}{4} \\ \square & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=-\frac {1}{y^{{3}/{2}} x} \\ {} & \circ & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & y^{\prime } y^{{3}/{2}}=-\frac {1}{x} \\ {} & \circ & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int y^{\prime } y^{{3}/{2}}d x =\int -\frac {1}{x}d x +\textit {\_C1} \\ {} & \circ & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \frac {2 y^{{5}/{2}}}{5}=-\ln \left (x \right )+\textit {\_C1} \\ {} & \circ & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\frac {\left (-80 \ln \left (x \right )+80 \textit {\_C1} \right )^{{2}/{5}}}{4} \\ \bullet & {} & \textrm {Set of solutions}\hspace {3pt} \\ {} & {} & \left \{y=\frac {\left (-80 \ln \left (x \right )+80 \mathit {C1} \right )^{{2}/{5}}}{4}, y=\frac {\left (80 \ln \left (x \right )+80 \mathit {C1} \right )^{{2}/{5}}}{4}\right \} \end {array} \]

1.57.3 Maple trace

Methods for first order ODEs:
 

1.57.4 Maple dsolve solution

Solving time : 0.042 (sec)
Leaf size : 29

dsolve(diff(y(x),x)^2 = 1/x^2/y(x)^3, 
       y(x),singsol=all)
 

\begin{align*} \ln \left (x \right )-\frac {2 y^{{5}/{2}}}{5}-c_1 &= 0 \\ \ln \left (x \right )+\frac {2 y^{{5}/{2}}}{5}-c_1 &= 0 \\ \end{align*}

1.57.5 Mathematica DSolve solution

Solving time : 0.132 (sec)
Leaf size : 45

DSolve[{(D[y[x],x])^2==1/(x^2*y[x]^3),{}}, 
       y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to \left (\frac {5}{2}\right )^{2/5} (-\log (x)+c_1){}^{2/5} \\ y(x)\to \left (\frac {5}{2}\right )^{2/5} (\log (x)+c_1){}^{2/5} \\ \end{align*}