3.26 problem 26

Internal problem ID [7216]
Internal file name [OUTPUT/6202_Sunday_June_05_2022_04_32_00_PM_14548145/index.tex]

Book: Own collection of miscellaneous problems
Section: section 3.0
Problem number: 26.
ODE order: 2.
ODE degree: 1.

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

Maple gives the following as the ode type

[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

\[ \boxed {y^{\prime \prime }+\sin \left (y\right ) {y^{\prime }}^{2}=0} \]

3.26.1 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 )+\sin \left (y \right ) p \left (y \right )^{2} = 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)\\ &= -p \sin \left (y \right ) \end {align*}

Where \(f(y)=-\sin \left (y \right )\) and \(g(p)=p\). Integrating both sides gives \begin {align*} \frac {1}{p} \,dp &= -\sin \left (y \right ) \,d y\\ \int { \frac {1}{p} \,dp} &= \int {-\sin \left (y \right ) \,d y}\\ \ln \left (p \right )&=\cos \left (y \right )+c_{1}\\ p&={\mathrm e}^{\cos \left (y \right )+c_{1}}\\ &=c_{1} {\mathrm e}^{\cos \left (y \right )} \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 } = c_{1} {\mathrm e}^{\cos \left (y\right )} \end {align*}

Integrating both sides gives \begin {align*} \int _{}^{y}\frac {{\mathrm e}^{-\cos \left (\textit {\_a} \right )}}{c_{1}}d \textit {\_a} = x +c_{2} \end {align*}

Summary

The solution(s) found are the following \begin{align*} \tag{1} \int _{}^{y}\frac {{\mathrm e}^{-\cos \left (\textit {\_a} \right )}}{c_{1}}d \textit {\_a} &= x +c_{2} \\ \end{align*}

Verification of solutions

\[ \int _{}^{y}\frac {{\mathrm e}^{-\cos \left (\textit {\_a} \right )}}{c_{1}}d \textit {\_a} = x +c_{2} \] Verified OK.

Maple trace

`Methods for second order ODEs: 
--- Trying classification methods --- 
trying 2nd order Liouville 
<- 2nd_order Liouville successful`
 

✓ Solution by Maple

Time used: 0.0 (sec). Leaf size: 21

dsolve(diff(y(x),x$2)+sin(y(x))*diff(y(x),x)^2=0,y(x), singsol=all)
 

\[ \int _{}^{y \left (x \right )}{\mathrm e}^{-\cos \left (\textit {\_a} \right )}d \textit {\_a} -c_{1} x -c_{2} = 0 \]

✓ Solution by Mathematica

Time used: 1.584 (sec). Leaf size: 111

DSolve[y''[x]+y[x]*Sin[y[x]](y'[x])^2==0,y[x],x,IncludeSingularSolutions -> True]
                                                                                    
                                                                                    
 

\begin{align*} y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {e^{\sin (K[1])-\cos (K[1]) K[1]}}{c_1}dK[1]\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}-\frac {e^{\sin (K[1])-\cos (K[1]) K[1]}}{c_1}dK[1]\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {e^{\sin (K[1])-\cos (K[1]) K[1]}}{c_1}dK[1]\&\right ][x+c_2] \\ \end{align*}