Internal problem ID [10088]
Internal file name [OUTPUT/9035_Monday_June_06_2022_06_15_40_AM_84296539/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1767 (book 6.176).
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]]
\[ \boxed {x y y^{\prime \prime }-4 x {y^{\prime }}^{2}+4 y y^{\prime }=0} \]
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 {4}{x}\\ g(y) &= -\frac {4}{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 \(g=-\frac {4}{y}\) and \(f=\frac {4}{x}\), then \begin {align*} \int -g d y &= \int \frac {4}{y}d y\\ &= 4 \ln \left (y\right )\\ \int -f d x &= \int -\frac {4}{x}d x\\ &= -4 \ln \left (x \right ) \end {align*}
Substituting the above into Eq(6A) gives \[ y^{\prime } = \frac {c_{2} y^{4}}{x^{4}} \] 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)\\ &= \frac {c_{2} y^{4}}{x^{4}} \end {align*}
Where \(f(x)=\frac {c_{2}}{x^{4}}\) and \(g(y)=y^{4}\). Integrating both sides gives \begin{align*} \frac {1}{y^{4}} \,dy &= \frac {c_{2}}{x^{4}} \,d x \\ \int { \frac {1}{y^{4}} \,dy} &= \int {\frac {c_{2}}{x^{4}} \,d x} \\ -\frac {1}{3 y^{3}}&=-\frac {c_{2}}{3 x^{3}}+c_{3} \\ \end{align*} The solution is \[ -\frac {1}{3 y^{3}}+\frac {c_{2}}{3 x^{3}}-c_{3} = 0 \]
The solution(s) found are the following \begin{align*} \tag{1} -\frac {1}{3 y^{3}}+\frac {c_{2}}{3 x^{3}}-c_{3} &= 0 \\ \end{align*}
Verification of solutions
\[ -\frac {1}{3 y^{3}}+\frac {c_{2}}{3 x^{3}}-c_{3} = 0 \] 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: 0.047 (sec). Leaf size: 68
dsolve(x*y(x)*diff(diff(y(x),x),x)-4*x*diff(y(x),x)^2+4*y(x)*diff(y(x),x)=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= 0 \\ y \left (x \right ) &= \frac {x}{\left (-3 c_{2} x^{3}+c_{1} \right )^{\frac {1}{3}}} \\ y \left (x \right ) &= -\frac {\left (1+i \sqrt {3}\right ) x}{2 \left (-3 c_{2} x^{3}+c_{1} \right )^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {\left (i \sqrt {3}-1\right ) x}{2 \left (-3 c_{2} x^{3}+c_{1} \right )^{\frac {1}{3}}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.67 (sec). Leaf size: 26
DSolve[4*y[x]*y'[x] - 4*x*y'[x]^2 + x*y[x]*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {c_2 x}{\sqrt [3]{1+c_1 x^3}} \\ y(x)\to 0 \\ \end{align*}