3.26 problem 26
Internal
problem
ID
[7864]
Book
:
Own
collection
of
miscellaneous
problems
Section
:
section
3.0
Problem
number
:
26
Date
solved
:
Monday, October 21, 2024 at 04:30:23 PM
CAS
classification
:
[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]
Solve
\begin{align*} y^{\prime \prime }+\sin \left (y\right ) {y^{\prime }}^{2}&=0 \end{align*}
3.26.1 Solved as second order missing x ode
Time used: 0.250 (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 )+\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 a linear first order is
\begin{align*} p^{\prime } + q(y)p &= p(y) \end{align*}
Comparing the above to the given ode shows that
\begin{align*} q(y) &=\sin \left (y \right )\\ p(y) &=0 \end{align*}
The integrating factor \(\mu \) is
\begin{align*} \mu &= e^{\int {q\,dy}}\\ &= {\mathrm e}^{\int \sin \left (y \right )d y}\\ &= {\mathrm e}^{-\cos \left (y \right )} \end{align*}
The ode becomes
\begin{align*} \frac {\mathop {\mathrm {d}}}{ \mathop {\mathrm {d}y}} \mu p &= 0 \\ \frac {\mathop {\mathrm {d}}}{ \mathop {\mathrm {d}y}} \left (p \,{\mathrm e}^{-\cos \left (y \right )}\right ) &= 0 \end{align*}
Integrating gives
\begin{align*} p \,{\mathrm e}^{-\cos \left (y \right )}&= \int {0 \,dy} + c_1 \\ &=c_1 \end{align*}
Dividing throughout by the integrating factor \({\mathrm e}^{-\cos \left (y \right )}\) gives the final solution
\[ p = {\mathrm e}^{\cos \left (y \right )} c_1 \]
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 } = {\mathrm e}^{\cos \left (y\right )} c_1 \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
\[ \int _{}^{y}\frac {{\mathrm e}^{-\cos \left (\tau \right )}}{c_1}d \tau = x +c_2 \]
Will add steps
showing solving for IC soon.
3.26.2 Maple step by step solution
3.26.3 Maple trace
Methods for second order ODEs:
3.26.4 Maple dsolve solution
Solving time : 0.003
(sec)
Leaf size : 21
dsolve(diff(diff(y(x),x),x)+sin(y(x))*diff(y(x),x)^2 = 0,
y(x),singsol=all)
\[
\int _{}^{y}{\mathrm e}^{-\cos \left (\textit {\_a} \right )}d \textit {\_a} -c_1 x -c_2 = 0
\]
3.26.5 Mathematica DSolve solution
Solving time : 1.302
(sec)
Leaf size : 111
DSolve[{D[y[x],{x,2}]+y[x]*Sin[y[x]](D[y[x],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*}