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2.1.52 problem 52

Maple step by step solution
Maple trace
Maple dsolve solution
Mathematica DSolve solution

Internal problem ID [8535]
Book : Second order enumerated odes
Section : section 1
Problem number : 52
Date solved : Sunday, November 10, 2024 at 04:00:56 AM
CAS classification : [[_2nd_order, _missing_x]]

Solve

\begin{align*} y {y^{\prime \prime }}^{3}+y^{3} {y^{\prime }}^{5}&=0 \end{align*}

Factoring the ode gives these factors

\begin{align*} \tag{1} y &= 0 \\ \tag{2} {y^{\prime }}^{5} y^{2} &= 0 \\ \end{align*}

Now each of the above equations is solved in turn.

Solving equation (1)

Solving for \(y\) from

\begin{align*} y = 0 \end{align*}

Solving gives \(y = 0\)

Solving equation (2)

Solving for the derivative gives these ODE’s to solve

\begin{align*} \tag{1} y^{\prime }&=0 \\ \tag{2} y^{\prime }&=0 \\ \tag{3} y^{\prime }&=0 \\ \tag{4} y^{\prime }&=0 \\ \tag{5} y^{\prime }&=0 \\ \end{align*}

Now each of the above is solved separately.

Solving Eq. (1)

Since the ode has the form \(y^{\prime }=f(x)\), then we only need to integrate \(f(x)\).

\begin{align*} \int {dy} &= \int {0\, dx} + c_1 \\ y &= c_1 \end{align*}

Solving Eq. (2)

Since the ode has the form \(y^{\prime }=f(x)\), then we only need to integrate \(f(x)\).

\begin{align*} \int {dy} &= \int {0\, dx} + c_2 \\ y &= c_2 \end{align*}

Solving Eq. (3)

Since the ode has the form \(y^{\prime }=f(x)\), then we only need to integrate \(f(x)\).

\begin{align*} \int {dy} &= \int {0\, dx} + c_3 \\ y &= c_3 \end{align*}

Solving Eq. (4)

Since the ode has the form \(y^{\prime }=f(x)\), then we only need to integrate \(f(x)\).

\begin{align*} \int {dy} &= \int {0\, dx} + c_4 \\ y &= c_4 \end{align*}

Solving Eq. (5)

Since the ode has the form \(y^{\prime }=f(x)\), then we only need to integrate \(f(x)\).

\begin{align*} \int {dy} &= \int {0\, dx} + c_5 \\ y &= c_5 \end{align*}

Maple step by step solution

Maple trace
`Methods for second order ODEs: 
   *** Sublevel 2 *** 
   Methods for second order ODEs: 
   Successful isolation of d^2y/dx^2: 3 solutions were found. Trying to solve each resulting ODE. 
      *** Sublevel 3 *** 
      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^2*_b(_a)^2)^(1/3)*_b(_a) = 0, _b(_a), HINT = [[_a, 5*_b]]` 
         symmetry methods on request 
      `, `1st order, trying reduction of order with given symmetries:`[_a, 5*_b]
Maple dsolve solution

Solving time : 0.311 (sec)
Leaf size : 208

dsolve(y(x)*diff(diff(y(x),x),x)^3+y(x)^3*diff(y(x),x)^5 = 0, 
       y(x),singsol=all)
\begin{align*} y &= 0 \\ y &= c_{1} \\ \int _{}^{y}\frac {1}{\operatorname {RootOf}\left (5 \left (\int _{\textit {\_g}}^{\textit {\_Z}}\frac {1}{\textit {\_a} \left (-\textit {\_a}^{2} \textit {\_f}^{2}\right )^{{1}/{3}}-5 \textit {\_f}}d \textit {\_f} \right )-\ln \left (\textit {\_a}^{5}+125\right )+5 c_{1} \right )}d \textit {\_a} -x -c_{2} &= 0 \\ \int _{}^{y}\frac {1}{\operatorname {RootOf}\left (-i \ln \left (\textit {\_a}^{5}+125\right )+\sqrt {3}\, \ln \left (\textit {\_a}^{5}+125\right )+20 \left (\int _{\textit {\_g}}^{\textit {\_Z}}\frac {1}{2 i \textit {\_a} \left (-\textit {\_a}^{2} \textit {\_f}^{2}\right )^{{1}/{3}}+5 i \textit {\_f} +5 \sqrt {3}\, \textit {\_f}}d \textit {\_f} \right )-20 c_{1} \right )}d \textit {\_a} -x -c_{2} &= 0 \\ \int _{}^{y}\frac {1}{\operatorname {RootOf}\left (\sqrt {3}\, \ln \left (\textit {\_a}^{5}+125\right )+i \ln \left (\textit {\_a}^{5}+125\right )+20 \left (\int _{\textit {\_g}}^{\textit {\_Z}}\frac {1}{-2 i \textit {\_a} \left (-\textit {\_a}^{2} \textit {\_f}^{2}\right )^{{1}/{3}}+5 \sqrt {3}\, \textit {\_f} -5 i \textit {\_f}}d \textit {\_f} \right )-20 c_{1} \right )}d \textit {\_a} -x -c_{2} &= 0 \\ \end{align*}
Mathematica DSolve solution

Solving time : 26.332 (sec)
Leaf size : 449

DSolve[{y[x]*D[y[x],{x,2}]^3+y[x]^3*D[y[x],x]^5==0,{}}, 
       y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)\to 0 \\ y(x)\to \text {InverseFunction}\left [\frac {27 \text {$\#$1} \operatorname {Hypergeometric2F1}\left (\frac {3}{5},3,\frac {8}{5},\frac {3 \text {$\#$1}^{5/3}}{5 c_1}\right )}{c_1{}^3}\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\frac {27 \text {$\#$1} \operatorname {Hypergeometric2F1}\left (\frac {3}{5},3,\frac {8}{5},-\frac {3 i \left (-i+\sqrt {3}\right ) \text {$\#$1}^{5/3}}{10 c_1}\right )}{c_1{}^3}\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\frac {27 \text {$\#$1} \operatorname {Hypergeometric2F1}\left (\frac {3}{5},3,\frac {8}{5},\frac {3 i \left (i+\sqrt {3}\right ) \text {$\#$1}^{5/3}}{10 c_1}\right )}{c_1{}^3}\&\right ][x+c_2] \\ y(x)\to 0 \\ y(x)\to \text {InverseFunction}\left [\frac {27 \text {$\#$1} \operatorname {Hypergeometric2F1}\left (\frac {3}{5},3,\frac {8}{5},\frac {3 \text {$\#$1}^{5/3}}{5 (-c_1)}\right )}{(-c_1){}^3}\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\frac {27 \text {$\#$1} \operatorname {Hypergeometric2F1}\left (\frac {3}{5},3,\frac {8}{5},-\frac {3 i \left (-i+\sqrt {3}\right ) \text {$\#$1}^{5/3}}{10 (-c_1)}\right )}{(-c_1){}^3}\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\frac {27 \text {$\#$1} \operatorname {Hypergeometric2F1}\left (\frac {3}{5},3,\frac {8}{5},\frac {3 i \left (i+\sqrt {3}\right ) \text {$\#$1}^{5/3}}{10 (-c_1)}\right )}{(-c_1){}^3}\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\frac {27 \text {$\#$1} \operatorname {Hypergeometric2F1}\left (\frac {3}{5},3,\frac {8}{5},\frac {3 \text {$\#$1}^{5/3}}{5 c_1}\right )}{c_1{}^3}\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\frac {27 \text {$\#$1} \operatorname {Hypergeometric2F1}\left (\frac {3}{5},3,\frac {8}{5},-\frac {3 i \left (-i+\sqrt {3}\right ) \text {$\#$1}^{5/3}}{10 c_1}\right )}{c_1{}^3}\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\frac {27 \text {$\#$1} \operatorname {Hypergeometric2F1}\left (\frac {3}{5},3,\frac {8}{5},\frac {3 i \left (i+\sqrt {3}\right ) \text {$\#$1}^{5/3}}{10 c_1}\right )}{c_1{}^3}\&\right ][x+c_2] \\ \end{align*}

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