Internal problem ID [10087]
Internal file name [OUTPUT/9034_Monday_June_06_2022_06_15_35_AM_46884265/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1766 (book 6.175).
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, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]
\[ \boxed {x y y^{\prime \prime }-2 x {y^{\prime }}^{2}+a y y^{\prime }=0} \]
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) &= \frac {a}{x}\\ g(y) &= -\frac {2}{y} \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 \(g=-\frac {2}{y}\) and \(f=\frac {a}{x}\), then \begin {align*} \int -g d y &= \int \frac {2}{y}d y\\ &= 2 \ln \left (y\right )\\ \int -f d x &= \int -\frac {a}{x}d x\\ &= -a \ln \left (x \right ) \end {align*}
Substituting the above into Eq(6A) gives \[ y^{\prime } = c_{2} y^{2} {\mathrm e}^{-a \ln \left (x \right )} \] 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} y^{2} {\mathrm e}^{-a \ln \left (x \right )} \end {align*}
Where \(f(x)=c_{2} {\mathrm e}^{-a \ln \left (x \right )}\) and \(g(y)=y^{2}\). Integrating both sides gives \begin{align*} \frac {1}{y^{2}} \,dy &= c_{2} {\mathrm e}^{-a \ln \left (x \right )} \,d x \\ \int { \frac {1}{y^{2}} \,dy} &= \int {c_{2} {\mathrm e}^{-a \ln \left (x \right )} \,d x} \\ -\frac {1}{y}&=-\frac {x c_{2} {\mathrm e}^{-a \ln \left (x \right )}}{a -1}+c_{3} \\ \end{align*} Which simplifies to \[ -\frac {1}{y}+\frac {x c_{2} x^{-a}}{a -1}-c_{3} = 0 \] The solution is \[ -\frac {1}{y}+\frac {x c_{2} x^{-a}}{a -1}-c_{3} = 0 \]
The solution(s) found are the following \begin{align*} \tag{1} -\frac {1}{y}+\frac {x c_{2} x^{-a}}{a -1}-c_{3} &= 0 \\ \end{align*}
Verification of solutions
\[ -\frac {1}{y}+\frac {x c_{2} x^{-a}}{a -1}-c_{3} = 0 \] 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: 0.031 (sec). Leaf size: 31
dsolve(x*y(x)*diff(diff(y(x),x),x)-2*x*diff(y(x),x)^2+a*y(x)*diff(y(x),x)=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= 0 \\ y \left (x \right ) &= -\frac {\left (a -1\right ) x^{a}}{c_{2} x^{a} \left (a -1\right )-c_{1} x} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.845 (sec). Leaf size: 29
DSolve[a*y[x]*y'[x] - 2*x*y'[x]^2 + x*y[x]*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {c_2 x^a}{x+(a-1) c_1 x^a} \\ y(x)\to 0 \\ \end{align*}