problem 1

Internal problem ID [7294]
Internal ﬁle name [OUTPUT/6280_Sunday_June_05_2022_04_37_01_PM_50628687/index.tex]

Book: Own collection of miscellaneous problems
Section: section 5.0
Problem number: 1.
ODE order: 2.
ODE degree: 1.

The type(s) of ODE detected by this program : "second_order_ode_missing_x", "second_order_ode_can_be_made_integrable"

5.1.1 Solving as second order ode can be made integrable ode

Multiplying the ode by \(y^{\prime }\) gives \[ y^{\prime } y^{\prime \prime }-A y^{\frac {2}{3}} y^{\prime } = 0 \] Integrating the above w.r.t \(x\) gives \begin {align*} \int \left (y^{\prime } y^{\prime \prime }-A y^{\frac {2}{3}} y^{\prime }\right )d x &= 0 \\ \frac {{y^{\prime }}^{2}}{2}-\frac {3 A y^{\frac {5}{3}}}{5} = c_2 \end {align*}

Which is now solved for \(y\). Solving the given ode for \(y^{\prime }\) results in \(2\) diﬀerential equations to solve. Each one of these will generate a solution. The equations generated are \begin {align*} y^{\prime }&=\frac {\sqrt {30 A y^{\frac {5}{3}}+50 c_{1}}}{5} \tag {1} \\ y^{\prime }&=-\frac {\sqrt {30 A y^{\frac {5}{3}}+50 c_{1}}}{5} \tag {2} \end {align*}

Integrating both sides gives \begin {align*} \int _{}^{y}\frac {5}{\sqrt {30 A \,\textit {\_a}^{\frac {5}{3}}+50 c_{1}}}d \textit {\_a} = x +c_{2} \end {align*}

Integrating both sides gives \begin {align*} \int _{}^{y}-\frac {5}{\sqrt {30 A \,\textit {\_a}^{\frac {5}{3}}+50 c_{1}}}d \textit {\_a} = x +c_{3} \end {align*}

Summary

The solution(s) found are the following \begin{align*} \tag{1} \int _{}^{y}\frac {5}{\sqrt {30 A \,\textit {\_a}^{\frac {5}{3}}+50 c_{1}}}d \textit {\_a} &= x +c_{2} \\ \tag{2} \int _{}^{y}-\frac {5}{\sqrt {30 A \,\textit {\_a}^{\frac {5}{3}}+50 c_{1}}}d \textit {\_a} &= x +c_{3} \\ \end{align*}

Veriﬁcation of solutions

\[ \int _{}^{y}\frac {5}{\sqrt {30 A \,\textit {\_a}^{\frac {5}{3}}+50 c_{1}}}d \textit {\_a} = x +c_{2} \] Veriﬁed OK.

\[ \int _{}^{y}-\frac {5}{\sqrt {30 A \,\textit {\_a}^{\frac {5}{3}}+50 c_{1}}}d \textit {\_a} = x +c_{3} \] Veriﬁed OK.

5.1.2 Solving as second order ode missing x ode

This is missing independent variable second order ode. Solved by reduction of order by using substitution which makes the dependent variable \(y\) an independent variable. Using \begin {align*} y' &= p(y) \end {align*}

Then \begin {align*} y'' &= \frac {dp}{dx}\\ &= \frac {dy}{dx} \frac {dp}{dy}\\ &= p \frac {dp}{dy} \end {align*}

Hence the ode becomes \begin {align*} p \left (y \right ) \left (\frac {d}{d y}p \left (y \right )\right )-A \,y^{\frac {2}{3}} = 0 \end {align*}

Which is now solved as ﬁrst order ode for \(p(y)\). In canonical form the ODE is \begin {align*} p' &= F(y,p)\\ &= f( y) g(p)\\ &= \frac {A \,y^{\frac {2}{3}}}{p} \end {align*}

Where \(f(y)=A \,y^{\frac {2}{3}}\) and \(g(p)=\frac {1}{p}\). Integrating both sides gives \begin{align*} \frac {1}{\frac {1}{p}} \,dp &= A \,y^{\frac {2}{3}} \,d y \\ \int { \frac {1}{\frac {1}{p}} \,dp} &= \int {A \,y^{\frac {2}{3}} \,d y} \\ \frac {p^{2}}{2}&=\frac {3 A \,y^{\frac {5}{3}}}{5}+c_{1} \\ \end{align*} The solution is \[ \frac {p \left (y \right )^{2}}{2}-\frac {3 A \,y^{\frac {5}{3}}}{5}-c_{1} = 0 \] For solution (1) found earlier, since \(p=y^{\prime }\) then we now have a new ﬁrst order ode to solve which is \begin {align*} \frac {{y^{\prime }}^{2}}{2}-\frac {3 A y^{\frac {5}{3}}}{5}-c_{1} = 0 \end {align*}

Solving the given ode for \(y^{\prime }\) results in \(2\) diﬀerential equations to solve. Each one of these will generate a solution. The equations generated are \begin {align*} y^{\prime }&=\frac {\sqrt {30 A y^{\frac {5}{3}}+50 c_{1}}}{5} \tag {1} \\ y^{\prime }&=-\frac {\sqrt {30 A y^{\frac {5}{3}}+50 c_{1}}}{5} \tag {2} \end {align*}

Integrating both sides gives \begin {align*} \int _{}^{y}\frac {5}{\sqrt {30 A \,\textit {\_a}^{\frac {5}{3}}+50 c_{1}}}d \textit {\_a} = x +c_{2} \end {align*}

Integrating both sides gives \begin {align*} \int _{}^{y}-\frac {5}{\sqrt {30 A \,\textit {\_a}^{\frac {5}{3}}+50 c_{1}}}d \textit {\_a} = x +c_{3} \end {align*}

Summary

Veriﬁcation of solutions

\[ \int _{}^{y}\frac {5}{\sqrt {30 A \,\textit {\_a}^{\frac {5}{3}}+50 c_{1}}}d \textit {\_a} = x +c_{2} \] Veriﬁed OK.

\[ \int _{}^{y}-\frac {5}{\sqrt {30 A \,\textit {\_a}^{\frac {5}{3}}+50 c_{1}}}d \textit {\_a} = x +c_{3} \] Veriﬁed OK.

Maple trace

`Methods for second order ODEs: 
--- Trying classification methods --- 
trying 2nd order Liouville 
trying 2nd order WeierstrassP 
trying 2nd order JacobiSN 
differential order: 2; trying a linearization to 3rd order 
trying 2nd order ODE linearizable_by_differentiation 
trying 2nd order, 2 integrating factors of the form mu(x,y) 
trying differential order: 2; missing variables 
`, `-> Computing symmetries using: way = 3 
-> Calling odsolve with the ODE`, (diff(_b(_a), _a))*_b(_a)-A*_a^(2/3) = 0, _b(_a), HINT = [[_a, (5/6)*_b]]`   *** Sublevel 2 *** 
   symmetry methods on request 
`, `1st order, trying reduction of order with given symmetries:`[_a, 5/6*_b]

\begin{align*} y \left (x \right ) &= 0 \\ -5 \left (\int _{}^{y \left (x \right )}\frac {1}{\sqrt {30 \textit {\_a}^{\frac {5}{3}} A -5 c_{1}}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ 5 \left (\int _{}^{y \left (x \right )}\frac {1}{\sqrt {30 \textit {\_a}^{\frac {5}{3}} A -5 c_{1}}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ \end{align*}

\[ \text {Solve}\left [\frac {y(x)^2 \left (1+\frac {6 A y(x)^{5/3}}{5 c_1}\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{5},\frac {8}{5},-\frac {6 A y(x)^{5/3}}{5 c_1}\right ){}^2}{\frac {6}{5} A y(x)^{5/3}+c_1}=(x+c_2){}^2,y(x)\right ] \]

5.1 problem 1

5.1.1 Solving as second order ode can be made integrable ode

5.1.2 Solving as second order ode missing x ode