Internal problem ID [7452]
Internal file name [OUTPUT/6419_Sunday_June_05_2022_04_51_38_PM_58438087/index.tex]
Book: Second order enumerated odes
Section: section 2
Problem number: 12.
ODE order: 2.
ODE degree: 1.
The type(s) of ODE detected by this program : "second_order_nonlinear_solved_by_mainardi_lioville_method"
Maple gives the following as the ode type
[_Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]
\[ \boxed {3 y^{\prime \prime }+\cos \left (x \right ) y^{\prime }+\sin \left (y\right ) {y^{\prime }}^{2}=0} \]
The ode has the Liouville form given by \begin {align*} y^{\prime \prime }+ f(x) y^{\prime } + g(y) {y^{\prime }}^{2} &= 0 \tag {1A} \end {align*}
Where in this problem \begin {align*} f(x) &= \frac {\cos \left (x \right )}{3}\\ g(y) &= \frac {\sin \left (y \right )}{3} \end {align*}
Dividing through by \(y^{\prime }\) then Eq (1A) becomes \begin {align*} \frac {y^{\prime \prime }}{y^{\prime }}+ f + g y^{\prime } &= 0 \tag {2A} \end {align*}
But the first term in Eq (2A) can be written as \begin {align*} \frac {y^{\prime \prime }}{y^{\prime }}&= \frac {d}{dx} \ln \left ( y^{\prime } \right )\tag {3A} \end {align*}
And the last term in Eq (2A) can be written as \begin {align*} g \frac {dy}{dx}&= \left ( \frac {d}{dy} \int g d y\right ) \frac {dy}{dx} \\ &= \frac {d}{dx} \int g d y\tag {4A} \end {align*}
Substituting (3A,4A) back into (2A) gives \begin {align*} \frac {d}{dx} \ln \left ( y^{\prime } \right ) + \frac {d}{dx} \int g d y &= -f \tag {5A} \end {align*}
Integrating the above w.r.t. \(x\) gives \begin {align*} \ln \left ( y^{\prime } \right ) + \int g d y &= - \int f d x + c_{1} \end {align*}
Where \(c_1\) is arbitrary constant. Taking the exponential of the above gives \begin {align*} y^{\prime } &= c_{2} e^{\int -g d y}\, e^{\int -f d x}\tag {6A} \end {align*}
Where \(c_{2}\) is a new arbitrary constant. But since \(g=\frac {\sin \left (y \right )}{3}\) and \(f=\frac {\cos \left (x \right )}{3}\), then \begin {align*} \int -g d y &= \int -\frac {\sin \left (y \right )}{3}d y\\ &= \frac {\cos \left (y\right )}{3}\\ \int -f d x &= \int -\frac {\cos \left (x \right )}{3}d x\\ &= -\frac {\sin \left (x \right )}{3} \end {align*}
Substituting the above into Eq(6A) gives \[ y^{\prime } = c_{2} {\mathrm e}^{\frac {\cos \left (y\right )}{3}} {\mathrm e}^{-\frac {\sin \left (x \right )}{3}} \] Which is now solved as first order separable ode. In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= f( x) g(y)\\ &= c_{2} {\mathrm e}^{\frac {\cos \left (y \right )}{3}} {\mathrm e}^{-\frac {\sin \left (x \right )}{3}} \end {align*}
Where \(f(x)=c_{2} {\mathrm e}^{-\frac {\sin \left (x \right )}{3}}\) and \(g(y)={\mathrm e}^{\frac {\cos \left (y \right )}{3}}\). Integrating both sides gives \begin{align*} \frac {1}{{\mathrm e}^{\frac {\cos \left (y \right )}{3}}} \,dy &= c_{2} {\mathrm e}^{-\frac {\sin \left (x \right )}{3}} \,d x \\ \int { \frac {1}{{\mathrm e}^{\frac {\cos \left (y \right )}{3}}} \,dy} &= \int {c_{2} {\mathrm e}^{-\frac {\sin \left (x \right )}{3}} \,d x} \\ \int _{}^{y}{\mathrm e}^{-\frac {\cos \left (\textit {\_a} \right )}{3}}d \textit {\_a}&=\int c_{2} {\mathrm e}^{-\frac {\sin \left (x \right )}{3}}d x +c_{3} \\ \end{align*} Which results in \[ \int _{}^{y}{\mathrm e}^{-\frac {\cos \left (\textit {\_a} \right )}{3}}d \textit {\_a}=\int c_{2} {\mathrm e}^{-\frac {\sin \left (x \right )}{3}}d x +c_{3} \] The solution is \[ \int _{}^{y}{\mathrm e}^{-\frac {\cos \left (\textit {\_a} \right )}{3}}d \textit {\_a} -\left (\int c_{2} {\mathrm e}^{-\frac {\sin \left (x \right )}{3}}d x \right )-c_{3} = 0 \]
The solution(s) found are the following \begin{align*} \tag{1} \int _{}^{y}{\mathrm e}^{-\frac {\cos \left (\textit {\_a} \right )}{3}}d \textit {\_a} -\left (\int c_{2} {\mathrm e}^{-\frac {\sin \left (x \right )}{3}}d x \right )-c_{3} &= 0 \\ \end{align*}
Verification of solutions
\[ \int _{}^{y}{\mathrm e}^{-\frac {\cos \left (\textit {\_a} \right )}{3}}d \textit {\_a} -\left (\int c_{2} {\mathrm e}^{-\frac {\sin \left (x \right )}{3}}d x \right )-c_{3} = 0 \] 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: 27
dsolve(3*diff(y(x),x$2)+cos(x)*diff(y(x),x)+sin(y(x))*(diff(y(x),x))^2=0,y(x), singsol=all)
\[ \int _{}^{y \left (x \right )}{\mathrm e}^{-\frac {\cos \left (\textit {\_a} \right )}{3}}d \textit {\_a} -c_{1} \left (\int {\mathrm e}^{-\frac {\sin \left (x \right )}{3}}d x \right )-c_{2} = 0 \]
✓ Solution by Mathematica
Time used: 0.601 (sec). Leaf size: 47
DSolve[3*y''[x]+Cos[x]*y'[x]+Sin[y[x]]*(y'[x])^2==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}e^{-\frac {1}{3} \cos (K[1])}dK[1]\&\right ]\left [\int _1^x-e^{-\frac {1}{3} \sin (K[2])} c_1dK[2]+c_2\right ] \]