1.58 problem 58

Internal problem ID [8022]
Book : First order enumerated odes
Section : section 1
Problem number : 58
Date solved : Monday, October 21, 2024 at 04:41:08 PM
CAS classification : [[_homogeneous, `class G`], _rational]

Solve

\begin{align*} {y^{\prime }}^{4}&=\frac {1}{x y^{3}} \end{align*}

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

Time used: 0.668 (sec)

The ode has the form

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

Where \(n=4, m=1, f=\frac {1}{x} , g=\frac {1}{y^{3}}\). Hence the ode is

\begin{align*} (y')^{4} &= \frac {1}{x \,y^{3}} \end{align*}

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

\begin{align*} y^{\prime } &=\left (f g \right )^{{1}/{4}}\\ y^{\prime } &=i \left (f g \right )^{{1}/{4}}\\ y^{\prime } &=-\left (f g \right )^{{1}/{4}}\\ y^{\prime } &=-i \left (f g \right )^{{1}/{4}} \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^{3}} &> 0 \end{align*}

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

\begin{align*} y^{\prime } &=f^{{1}/{4}} g^{{1}/{4}}\\ y^{\prime } &=i f^{{1}/{4}} g^{{1}/{4}}\\ y^{\prime } &=-f^{{1}/{4}} g^{{1}/{4}}\\ y^{\prime } &=-i f^{{1}/{4}} g^{{1}/{4}} \end{align*}

Therefore

\begin{align*} \frac {1}{g^{{1}/{4}}} \, dy &= \left (f^{{1}/{4}}\right )\,dx\\ -\frac {i}{g^{{1}/{4}}} \, dy &= \left (f^{{1}/{4}}\right )\,dx\\ -\frac {1}{g^{{1}/{4}}} \, dy &= \left (f^{{1}/{4}}\right )\,dx\\ \frac {i}{g^{{1}/{4}}} \, dy &= \left (f^{{1}/{4}}\right )\,dx \end{align*}

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

\begin{align*} \frac {1}{\left (\frac {1}{y^{3}}\right )^{{1}/{4}}} \, dy &= \left (\left (\frac {1}{x}\right )^{{1}/{4}}\right )\,dx\\ -\frac {i}{\left (\frac {1}{y^{3}}\right )^{{1}/{4}}} \, dy &= \left (\left (\frac {1}{x}\right )^{{1}/{4}}\right )\,dx\\ -\frac {1}{\left (\frac {1}{y^{3}}\right )^{{1}/{4}}} \, dy &= \left (\left (\frac {1}{x}\right )^{{1}/{4}}\right )\,dx\\ \frac {i}{\left (\frac {1}{y^{3}}\right )^{{1}/{4}}} \, dy &= \left (\left (\frac {1}{x}\right )^{{1}/{4}}\right )\,dx \end{align*}

Integrating now gives the following solutions

\begin{align*} \int \frac {1}{\left (\frac {1}{y^{3}}\right )^{{1}/{4}}}d y &= \int \left (\frac {1}{x}\right )^{{1}/{4}}d x +c_1\\ \frac {4 y^{4} \left (\frac {1}{y^{3}}\right )^{{3}/{4}}}{7} &= \frac {4 x \left (\frac {1}{x}\right )^{{1}/{4}}}{3}\\ \int -\frac {i}{\left (\frac {1}{y^{3}}\right )^{{1}/{4}}}d y &= \int \left (\frac {1}{x}\right )^{{1}/{4}}d x +c_1\\ -\frac {4 i y^{4} \left (\frac {1}{y^{3}}\right )^{{3}/{4}}}{7} &= \frac {4 x \left (\frac {1}{x}\right )^{{1}/{4}}}{3}\\ \int -\frac {1}{\left (\frac {1}{y^{3}}\right )^{{1}/{4}}}d y &= \int \left (\frac {1}{x}\right )^{{1}/{4}}d x +c_1\\ -\frac {4 y^{4} \left (\frac {1}{y^{3}}\right )^{{3}/{4}}}{7} &= \frac {4 x \left (\frac {1}{x}\right )^{{1}/{4}}}{3}\\ \int \frac {i}{\left (\frac {1}{y^{3}}\right )^{{1}/{4}}}d y &= \int \left (\frac {1}{x}\right )^{{1}/{4}}d x +c_1\\ \frac {4 i y^{4} \left (\frac {1}{y^{3}}\right )^{{3}/{4}}}{7} &= \frac {4 x \left (\frac {1}{x}\right )^{{1}/{4}}}{3} \end{align*}

Therefore

\begin{align*} \frac {4 y^{4} \left (\frac {1}{y^{3}}\right )^{{3}/{4}}}{7} &= \frac {4 x \left (\frac {1}{x}\right )^{{1}/{4}}}{3}+c_1 \\ -\frac {4 i y^{4} \left (\frac {1}{y^{3}}\right )^{{3}/{4}}}{7} &= \frac {4 x \left (\frac {1}{x}\right )^{{1}/{4}}}{3}+c_1 \\ -\frac {4 y^{4} \left (\frac {1}{y^{3}}\right )^{{3}/{4}}}{7} &= \frac {4 x \left (\frac {1}{x}\right )^{{1}/{4}}}{3}+c_1 \\ \frac {4 i y^{4} \left (\frac {1}{y^{3}}\right )^{{3}/{4}}}{7} &= \frac {4 x \left (\frac {1}{x}\right )^{{1}/{4}}}{3}+c_1 \\ \end{align*}

1.58.2 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & {y^{\prime }}^{4}=\frac {1}{x y^{3}} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \left [y^{\prime }=\frac {\left (x^{3} y\right )^{{1}/{4}}}{x y}, y^{\prime }=-\frac {\left (x^{3} y\right )^{{1}/{4}}}{x y}, y^{\prime }=\frac {\mathrm {-I} \left (x^{3} y\right )^{{1}/{4}}}{x y}, y^{\prime }=\frac {\mathrm {I} \left (x^{3} y\right )^{{1}/{4}}}{x y}\right ] \\ \bullet & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=\frac {\left (x^{3} y\right )^{{1}/{4}}}{x y} \\ \bullet & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=-\frac {\left (x^{3} y\right )^{{1}/{4}}}{x y} \\ \bullet & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=\frac {\mathrm {-I} \left (x^{3} y\right )^{{1}/{4}}}{x y} \\ \bullet & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=\frac {\mathrm {I} \left (x^{3} y\right )^{{1}/{4}}}{x y} \\ \bullet & {} & \textrm {Set of solutions}\hspace {3pt} \\ {} & {} & \left \{\mathit {workingODE} , \mathit {workingODE} , \mathit {workingODE} , \mathit {workingODE}\right \} \end {array} \]

1.58.3 Maple trace

Methods for first order ODEs:
 

1.58.4 Maple dsolve solution

Solving time : 0.054 (sec)
Leaf size : 121

dsolve(diff(y(x),x)^4 = 1/x/y(x)^3, 
       y(x),singsol=all)
 

\begin{align*} -\frac {7 x^{3}-3 y \left (x^{3} y\right )^{{3}/{4}}+c_1 \,x^{{9}/{4}}}{x^{{9}/{4}}} &= 0 \\ \frac {-7 x^{3}+3 i y \left (x^{3} y\right )^{{3}/{4}}-c_1 \,x^{{9}/{4}}}{x^{{9}/{4}}} &= 0 \\ \frac {7 x^{3}+3 i y \left (x^{3} y\right )^{{3}/{4}}-c_1 \,x^{{9}/{4}}}{x^{{9}/{4}}} &= 0 \\ \frac {7 x^{3}+3 y \left (x^{3} y\right )^{{3}/{4}}-c_1 \,x^{{9}/{4}}}{x^{{9}/{4}}} &= 0 \\ \end{align*}

1.58.5 Mathematica DSolve solution

Solving time : 6.693 (sec)
Leaf size : 129

DSolve[{(D[y[x],x])^4==1/(x*y[x]^3),{}}, 
       y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to \frac {\left (-\frac {28 x^{3/4}}{3}+7 c_1\right ){}^{4/7}}{2 \sqrt [7]{2}} \\ y(x)\to \frac {\left (7 c_1-\frac {28}{3} i x^{3/4}\right ){}^{4/7}}{2 \sqrt [7]{2}} \\ y(x)\to \frac {\left (\frac {28}{3} i x^{3/4}+7 c_1\right ){}^{4/7}}{2 \sqrt [7]{2}} \\ y(x)\to \frac {\left (\frac {28 x^{3/4}}{3}+7 c_1\right ){}^{4/7}}{2 \sqrt [7]{2}} \\ \end{align*}