2.6 problem 6

Internal problem ID [3145]
Internal file name [OUTPUT/2637_Sunday_June_05_2022_08_37_50_AM_9569087/index.tex]

Book: An introduction to the solution and applications of differential equations, J.W. Searl, 1966
Section: Chapter 4, Ex. 4.2
Problem number: 6.
ODE order: 1.
ODE degree: 1.

The type(s) of ODE detected by this program : "quadrature"

Maple gives the following as the ode type

[_quadrature]

\[ \boxed {y^{2} y^{\prime }-3 y^{6}=2} \] With initial conditions \begin {align*} [y \left (0\right ) = 0] \end {align*}

2.6.1 Existence and uniqueness analysis

This is non linear first order ODE. In canonical form it is written as \begin {align*} y^{\prime } &= f(x,y)\\ &= \frac {3 y^{6}+2}{y^{2}} \end {align*}

The \(y\) domain of \(f(x,y)\) when \(x=0\) is \[ \{y <0\boldsymbol {\lor }0

2.6.2 Solving as separable ode

In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= f( x) g(y)\\ &= \frac {3 y^{6}+2}{y^{2}} \end {align*}

Where \(f(x)=1\) and \(g(y)=\frac {3 y^{6}+2}{y^{2}}\). Integrating both sides gives \begin{align*} \frac {1}{\frac {3 y^{6}+2}{y^{2}}} \,dy &= 1 \,d x \\ \int { \frac {1}{\frac {3 y^{6}+2}{y^{2}}} \,dy} &= \int {1 \,d x} \\ \frac {\sqrt {6}\, \arctan \left (\frac {y^{3} \sqrt {6}}{2}\right )}{18}&=x +c_{1} \\ \end{align*} Which results in \begin{align*} y &= \frac {9^{\frac {1}{3}} \left (\sqrt {6}\, \tan \left (3 \left (x +c_{1} \right ) \sqrt {6}\right )\right )^{\frac {1}{3}}}{3} \\ y &= -\frac {9^{\frac {1}{3}} \left (\sqrt {6}\, \tan \left (3 \left (x +c_{1} \right ) \sqrt {6}\right )\right )^{\frac {1}{3}}}{6}+\frac {i \sqrt {3}\, 9^{\frac {1}{3}} \left (\sqrt {6}\, \tan \left (3 \left (x +c_{1} \right ) \sqrt {6}\right )\right )^{\frac {1}{3}}}{6} \\ y &= -\frac {9^{\frac {1}{3}} \left (\sqrt {6}\, \tan \left (3 \left (x +c_{1} \right ) \sqrt {6}\right )\right )^{\frac {1}{3}}}{6}-\frac {i \sqrt {3}\, 9^{\frac {1}{3}} \left (\sqrt {6}\, \tan \left (3 \left (x +c_{1} \right ) \sqrt {6}\right )\right )^{\frac {1}{3}}}{6} \\ \end{align*} Initial conditions are used to solve for \(c_{1}\). Substituting \(x=0\) and \(y=0\) in the above solution gives an equation to solve for the constant of integration. \begin {align*} 0 = -\frac {i \tan \left (3 c_{1} \sqrt {6}\right )^{\frac {1}{3}} 6^{\frac {1}{6}} 3^{\frac {1}{6}}}{2}-\frac {\tan \left (3 c_{1} \sqrt {6}\right )^{\frac {1}{3}} 6^{\frac {1}{6}} 3^{\frac {2}{3}}}{6} \end {align*}

The solutions are \begin {align*} c_{1} = 0 \end {align*}

Trying the constant \begin {align*} c_{1} = 0 \end {align*}

Substituting this in the general solution gives \begin {align*} y&=-\frac {i \tan \left (3 x \sqrt {6}\right )^{\frac {1}{3}} 6^{\frac {1}{6}} 3^{\frac {1}{6}}}{2}-\frac {\tan \left (3 x \sqrt {6}\right )^{\frac {1}{3}} 6^{\frac {1}{6}} 3^{\frac {2}{3}}}{6} \end {align*}

The constant \(c_{1} = 0\) gives valid solution.

Initial conditions are used to solve for \(c_{1}\). Substituting \(x=0\) and \(y=0\) in the above solution gives an equation to solve for the constant of integration. \begin {align*} 0 = \frac {i \tan \left (3 c_{1} \sqrt {6}\right )^{\frac {1}{3}} 6^{\frac {1}{6}} 3^{\frac {1}{6}}}{2}-\frac {\tan \left (3 c_{1} \sqrt {6}\right )^{\frac {1}{3}} 6^{\frac {1}{6}} 3^{\frac {2}{3}}}{6} \end {align*}

The solutions are \begin {align*} c_{1} = 0 \end {align*}

Trying the constant \begin {align*} c_{1} = 0 \end {align*}

Substituting this in the general solution gives \begin {align*} y&=\frac {i \tan \left (3 x \sqrt {6}\right )^{\frac {1}{3}} 6^{\frac {1}{6}} 3^{\frac {1}{6}}}{2}-\frac {\tan \left (3 x \sqrt {6}\right )^{\frac {1}{3}} 6^{\frac {1}{6}} 3^{\frac {2}{3}}}{6} \end {align*}

The constant \(c_{1} = 0\) gives valid solution.

Initial conditions are used to solve for \(c_{1}\). Substituting \(x=0\) and \(y=0\) in the above solution gives an equation to solve for the constant of integration. \begin {align*} 0 = \frac {\tan \left (3 c_{1} \sqrt {6}\right )^{\frac {1}{3}} 6^{\frac {1}{6}} 9^{\frac {1}{3}}}{3} \end {align*}

The solutions are \begin {align*} c_{1} = 0 \end {align*}

Trying the constant \begin {align*} c_{1} = 0 \end {align*}

Substituting this in the general solution gives \begin {align*} y&=\frac {\tan \left (3 x \sqrt {6}\right )^{\frac {1}{3}} 6^{\frac {1}{6}} 9^{\frac {1}{3}}}{3} \end {align*}

The constant \(c_{1} = 0\) gives valid solution.

Summary

The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {\tan \left (3 x \sqrt {6}\right )^{\frac {1}{3}} 6^{\frac {1}{6}} 9^{\frac {1}{3}}}{3} \\ \tag{2} y &= \frac {i \tan \left (3 x \sqrt {6}\right )^{\frac {1}{3}} 6^{\frac {1}{6}} 3^{\frac {1}{6}}}{2}-\frac {\tan \left (3 x \sqrt {6}\right )^{\frac {1}{3}} 6^{\frac {1}{6}} 3^{\frac {2}{3}}}{6} \\ \tag{3} y &= -\frac {i \tan \left (3 x \sqrt {6}\right )^{\frac {1}{3}} 6^{\frac {1}{6}} 3^{\frac {1}{6}}}{2}-\frac {\tan \left (3 x \sqrt {6}\right )^{\frac {1}{3}} 6^{\frac {1}{6}} 3^{\frac {2}{3}}}{6} \\ \end{align*}

(a) Solution plot

(b) Slope field plot

Verification of solutions

\[ y = \frac {\tan \left (3 x \sqrt {6}\right )^{\frac {1}{3}} 6^{\frac {1}{6}} 9^{\frac {1}{3}}}{3} \] Verified OK.

\[ y = \frac {i \tan \left (3 x \sqrt {6}\right )^{\frac {1}{3}} 6^{\frac {1}{6}} 3^{\frac {1}{6}}}{2}-\frac {\tan \left (3 x \sqrt {6}\right )^{\frac {1}{3}} 6^{\frac {1}{6}} 3^{\frac {2}{3}}}{6} \] Verified OK.

\[ y = -\frac {i \tan \left (3 x \sqrt {6}\right )^{\frac {1}{3}} 6^{\frac {1}{6}} 3^{\frac {1}{6}}}{2}-\frac {\tan \left (3 x \sqrt {6}\right )^{\frac {1}{3}} 6^{\frac {1}{6}} 3^{\frac {2}{3}}}{6} \] Verified OK.

2.6.3 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left [y^{2} y^{\prime }-3 y^{6}=2, y \left (0\right )=0\right ] \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {2+3 y^{6}}{y^{2}} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime } y^{2}}{2+3 y^{6}}=1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime } y^{2}}{2+3 y^{6}}d x =\int 1d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \frac {\sqrt {6}\, \arctan \left (\frac {y^{3} \sqrt {6}}{2}\right )}{18}=x +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\frac {9^{\frac {1}{3}} \left (\sqrt {6}\, \tan \left (3 \left (x +c_{1} \right ) \sqrt {6}\right )\right )^{\frac {1}{3}}}{3} \\ \bullet & {} & \textrm {Use initial condition}\hspace {3pt} y \left (0\right )=0 \\ {} & {} & 0=\frac {9^{\frac {1}{3}} \left (\tan \left (3 c_{1} \sqrt {6}\right ) \sqrt {6}\right )^{\frac {1}{3}}}{3} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} c_{1} \\ {} & {} & c_{1} =0 \\ \bullet & {} & \textrm {Substitute}\hspace {3pt} c_{1} =0\hspace {3pt}\textrm {into general solution and simplify}\hspace {3pt} \\ {} & {} & y=\frac {3^{\frac {5}{6}} 2^{\frac {1}{6}} \tan \left (3 x \sqrt {6}\right )^{\frac {1}{3}}}{3} \\ \bullet & {} & \textrm {Solution to the IVP}\hspace {3pt} \\ {} & {} & y=\frac {3^{\frac {5}{6}} 2^{\frac {1}{6}} \tan \left (3 x \sqrt {6}\right )^{\frac {1}{3}}}{3} \end {array} \]

Maple trace

`Methods for first order ODEs: 
--- Trying classification methods --- 
trying a quadrature 
trying 1st order linear 
trying Bernoulli 
trying separable 
<- separable successful`
 

✓ Solution by Maple

Time used: 0.235 (sec). Leaf size: 77

dsolve([y(x)^2*diff(y(x),x)=2+3*y(x)^6,y(0) = 0],y(x), singsol=all)
 

\begin{align*} y \left (x \right ) &= \frac {3^{\frac {5}{6}} 2^{\frac {1}{6}} \tan \left (3 \sqrt {6}\, x \right )^{\frac {1}{3}}}{3} \\ y \left (x \right ) &= \frac {\tan \left (3 \sqrt {6}\, x \right )^{\frac {1}{3}} \left (3 i 3^{\frac {1}{6}}-3^{\frac {2}{3}}\right ) 6^{\frac {1}{6}}}{6} \\ y \left (x \right ) &= -\frac {\tan \left (3 \sqrt {6}\, x \right )^{\frac {1}{3}} \left (3 i 3^{\frac {1}{6}}+3^{\frac {2}{3}}\right ) 6^{\frac {1}{6}}}{6} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.017 (sec). Leaf size: 87

DSolve[{y[x]^2*y'[x]==2+3*y[x]^6,y[0]==0},y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \sqrt [6]{\frac {2}{3}} \sqrt [3]{\tan \left (3 \sqrt {6} x\right )} \\ y(x)\to -\sqrt [3]{-1} \sqrt [6]{\frac {2}{3}} \sqrt [3]{\tan \left (3 \sqrt {6} x\right )} \\ y(x)\to (-1)^{2/3} \sqrt [6]{\frac {2}{3}} \sqrt [3]{\tan \left (3 \sqrt {6} x\right )} \\ \end{align*}