Internal problem ID [9963]
Internal file name [OUTPUT/8910_Monday_June_06_2022_05_50_18_AM_6027267/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1641 (6.51).
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]]
\[ \boxed {y^{\prime \prime }+f \left (y\right ) {y^{\prime }}^{2}+g \left (x \right ) y^{\prime }=0} \]
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) &= g \left (x \right )\\ g(y) &= f \left (y \right ) \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 \(g=f \left (y \right )\) and \(f=g \left (x \right )\), then \begin {align*} \int -g d y &= \int -f \left (y \right )d y\\ &= -\left (\int f \left (y \right )d \textit {\_a} \right )\\ \int -f d x &= \int -g \left (x \right )d x\\ &= -\left (\int g \left (x \right )d x \right ) \end {align*}
Substituting the above into Eq(6A) gives \[ y^{\prime } = c_{2} {\mathrm e}^{-\left (\int f \left (y \right )d \textit {\_a} \right )} {\mathrm e}^{-\left (\int g \left (x \right )d x \right )} \] Which is now solved as first order separable ode. Integrating the above gives \[ \int _{0}^{y}{\mathrm e}^{\int f \left (y \right )d \textit {\_a}}d \textit {\_a} = c_{1} \left (\int {\mathrm e}^{-\left (\int g \left (x \right )d x \right )}d x \right )+c_{2} \]
The solution(s) found are the following \begin{align*} \tag{1} \int _{0}^{y}{\mathrm e}^{\int f \left (y \right )d \textit {\_a}}d \textit {\_a} &= c_{1} \left (\int {\mathrm e}^{-\left (\int g \left (x \right )d x \right )}d x \right )+c_{2} \\ \end{align*}
Verification of solutions
\[ \int _{0}^{y}{\mathrm e}^{\int f \left (y \right )d \textit {\_a}}d \textit {\_a} = c_{1} \left (\int {\mathrm e}^{-\left (\int g \left (x \right )d x \right )}d x \right )+c_{2} \] 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: 0.015 (sec). Leaf size: 29
dsolve(diff(diff(y(x),x),x)+f(y(x))*diff(y(x),x)^2+g(x)*diff(y(x),x)=0,y(x), singsol=all)
\[ \int _{}^{y \left (x \right )}{\mathrm e}^{\int f \left (\textit {\_b} \right )d \textit {\_b}}d \textit {\_b} -c_{1} \left (\int {\mathrm e}^{-\left (\int g \left (x \right )d x \right )}d x \right )-c_{2} = 0 \]
✓ Solution by Mathematica
Time used: 0.079 (sec). Leaf size: 61
DSolve[g[x]*y'[x] + f[y[x]]*y'[x]^2 + y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\exp \left (-\int _1^{K[1]}-f(K[1])dK[1]\right )dK[1]\&\right ]\left [\int _1^x-\exp \left (-\int _1^{K[2]}g(K[2])dK[2]\right ) c_1dK[2]+c_2\right ] \]