5.1 problem 1

Internal problem ID [7942]
Book : Own collection of miscellaneous problems
Section : section 5.0
Problem number : 1
Date solved : Monday, October 21, 2024 at 04:36:25 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

Solve

\begin{align*} y^{\prime \prime }&=A y^{{2}/{3}} \end{align*}

5.1.1 Solved as second order missing x ode

Time used: 87.043 (sec)

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 \end{align*}

Then

\begin{align*} y'' &= \frac {dp}{dx}\\ &= \frac {dp}{dy}\frac {dy}{dx}\\ &= 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^{{2}/{3}} \end{align*}

Which is now solved as first order ode for \(p(y)\).

The ode \(p^{\prime } = \frac {A \,y^{{2}/{3}}}{p}\) is separable as it can be written as

\begin{align*} p^{\prime }&= \frac {A \,y^{{2}/{3}}}{p}\\ &= f(y) g(p) \end{align*}

Where

\begin{align*} f(y) &= A \,y^{{2}/{3}}\\ g(p) &= \frac {1}{p} \end{align*}

Integrating gives

\begin{align*} \int { \frac {1}{g(p)} \,dp} &= \int { f(y) \,dy}\\ \int { p\,dp} &= \int { A \,y^{{2}/{3}} \,dy}\\ \frac {p^{2}}{2}&=\frac {3 y^{{5}/{3}} A}{5}+c_1 \end{align*}

Solving for \(p\) from the above solution(s) gives (after possible removing of solutions that do not verify)

\begin{align*} p&=-\frac {\sqrt {30 y^{{5}/{3}} A +50 c_1}}{5}\\ p&=\frac {\sqrt {30 y^{{5}/{3}} A +50 c_1}}{5} \end{align*}

For solution (1) found earlier, since \(p=y^{\prime }\) then we now have a new first order ode to solve which is

\begin{align*} y^{\prime } = -\frac {\sqrt {30 y^{{5}/{3}} A +50 c_1}}{5} \end{align*}

Since initial conditions \(\left (x_0,y_0\right ) \) are given, then the result can be written as Since unable to evaluate the integral, and no initial conditions are given, then the result becomes

\[ \int _{}^{y}-\frac {5}{\sqrt {30 \tau ^{{5}/{3}} A +50 c_1}}d \tau = x +c_2 \]

Singular solutions are found by solving

\begin{align*} -\frac {\sqrt {30 y^{{5}/{3}} A +50 c_1}}{5}&= 0 \end{align*}

for \(y\). This is because we had to divide by this in the above step. This gives the following singular solution(s), which also have to satisfy the given ODE.

\begin{align*} y = \frac {5^{{3}/{5}} 3^{{2}/{5}} \left (-c_1 \,A^{4}\right )^{{3}/{5}}}{3 A^{3}} \end{align*}

For solution (2) found earlier, since \(p=y^{\prime }\) then we now have a new first order ode to solve which is

\begin{align*} y^{\prime } = \frac {\sqrt {30 y^{{5}/{3}} A +50 c_1}}{5} \end{align*}

Since initial conditions \(\left (x_0,y_0\right ) \) are given, then the result can be written as Since unable to evaluate the integral, and no initial conditions are given, then the result becomes

\[ \int _{}^{y}\frac {5}{\sqrt {30 \tau ^{{5}/{3}} A +50 c_1}}d \tau = x +c_3 \]

Singular solutions are found by solving

\begin{align*} \frac {\sqrt {30 y^{{5}/{3}} A +50 c_1}}{5}&= 0 \end{align*}

for \(y\). This is because we had to divide by this in the above step. This gives the following singular solution(s), which also have to satisfy the given ODE.

\begin{align*} y = \frac {5^{{3}/{5}} 3^{{2}/{5}} \left (-c_1 \,A^{4}\right )^{{3}/{5}}}{3 A^{3}} \end{align*}

Will add steps showing solving for IC soon.

The solution

\[ y = \frac {5^{{3}/{5}} 3^{{2}/{5}} \left (-c_1 \,A^{4}\right )^{{3}/{5}}}{3 A^{3}} \]

was found not to satisfy the ode or the IC. Hence it is removed.

5.1.2 Solved as second order can be made integrable

Time used: 80.273 (sec)

Multiplying the ode by \(y^{\prime }\) gives

\[ -y^{{2}/{3}} A y^{\prime }+y^{\prime } y^{\prime \prime } = 0 \]

Integrating the above w.r.t \(x\) gives

\begin{align*} \int \left (-y^{{2}/{3}} A y^{\prime }+y^{\prime } y^{\prime \prime }\right )d x &= 0 \\ -\frac {3 y^{{5}/{3}} A}{5}+\frac {{y^{\prime }}^{2}}{2} &= c_1 \end{align*}

Which is now solved for \(y\). Solving for the derivative gives these ODE’s to solve

\begin{align*} \tag{1} y^{\prime }&=\frac {\sqrt {30 y^{{5}/{3}} A +50 c_1}}{5} \\ \tag{2} y^{\prime }&=-\frac {\sqrt {30 y^{{5}/{3}} A +50 c_1}}{5} \\ \end{align*}

Now each of the above is solved separately.

Solving Eq. (1)

Since initial conditions \(\left (x_0,y_0\right ) \) are given, then the result can be written as Since unable to evaluate the integral, and no initial conditions are given, then the result becomes

\[ \int _{}^{y}\frac {5}{\sqrt {30 \tau ^{{5}/{3}} A +50 c_1}}d \tau = x +c_2 \]

Singular solutions are found by solving

\begin{align*} \frac {\sqrt {30 y^{{5}/{3}} A +50 c_1}}{5}&= 0 \end{align*}

for \(y\). This is because we had to divide by this in the above step. This gives the following singular solution(s), which also have to satisfy the given ODE.

\begin{align*} y = \frac {5^{{3}/{5}} 3^{{2}/{5}} \left (-c_1 \,A^{4}\right )^{{3}/{5}}}{3 A^{3}} \end{align*}

Solving Eq. (2)

Since initial conditions \(\left (x_0,y_0\right ) \) are given, then the result can be written as Since unable to evaluate the integral, and no initial conditions are given, then the result becomes

\[ \int _{}^{y}-\frac {5}{\sqrt {30 \tau ^{{5}/{3}} A +50 c_1}}d \tau = x +c_3 \]

Singular solutions are found by solving

\begin{align*} -\frac {\sqrt {30 y^{{5}/{3}} A +50 c_1}}{5}&= 0 \end{align*}

for \(y\). This is because we had to divide by this in the above step. This gives the following singular solution(s), which also have to satisfy the given ODE.

\begin{align*} y = \frac {5^{{3}/{5}} 3^{{2}/{5}} \left (-c_1 \,A^{4}\right )^{{3}/{5}}}{3 A^{3}} \end{align*}

Will add steps showing solving for IC soon.

The solution

\[ y = \frac {5^{{3}/{5}} 3^{{2}/{5}} \left (-c_1 \,A^{4}\right )^{{3}/{5}}}{3 A^{3}} \]

was found not to satisfy the ode or the IC. Hence it is removed.

5.1.3 Maple step by step solution

5.1.4 Maple trace

Methods for second order ODEs:
 

5.1.5 Maple dsolve solution

Solving time : 0.036 (sec)
Leaf size : 61

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

\begin{align*} y &= 0 \\ -5 \left (\int _{}^{y}\frac {1}{\sqrt {30 \textit {\_a}^{{5}/{3}} A -5 c_1}}d \textit {\_a} \right )-x -c_2 &= 0 \\ 5 \left (\int _{}^{y}\frac {1}{\sqrt {30 \textit {\_a}^{{5}/{3}} A -5 c_1}}d \textit {\_a} \right )-x -c_2 &= 0 \\ \end{align*}

5.1.6 Mathematica DSolve solution

Solving time : 0.117 (sec)
Leaf size : 75

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

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