Internal problem ID [10076]
Internal file name [OUTPUT/9023_Monday_June_06_2022_06_14_15_AM_59033396/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1755 (book 6.164).
ODE order: 2.
ODE degree: 1.
The type(s) of ODE detected by this program : "second_order_ode_missing_x"
Maple gives the following as the ode type
[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]
\[ \boxed {n y y^{\prime \prime }-\left (n -1\right ) {y^{\prime }}^{2}=0} \]
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(y) \end {align*}
Then \begin {align*} y'' &= \frac {dp}{dx}\\ &= \frac {dy}{dx} \frac {dp}{dy}\\ &= p \frac {dp}{dy} \end {align*}
Hence the ode becomes \begin {align*} n y p \left (y \right ) \left (\frac {d}{d y}p \left (y \right )\right )+\left (-n p \left (y \right )+p \left (y \right )\right ) p \left (y \right ) = 0 \end {align*}
Which is now solved as first order ode for \(p(y)\). In canonical form the ODE is \begin {align*} p' &= F(y,p)\\ &= f( y) g(p)\\ &= \frac {p \left (n -1\right )}{n y} \end {align*}
Where \(f(y)=\frac {n -1}{y n}\) and \(g(p)=p\). Integrating both sides gives \begin {align*} \frac {1}{p} \,dp &= \frac {n -1}{y n} \,d y\\ \int { \frac {1}{p} \,dp} &= \int {\frac {n -1}{y n} \,d y}\\ \ln \left (p \right )&=\frac {\left (n -1\right ) \ln \left (y \right )}{n}+c_{1}\\ p&={\mathrm e}^{\frac {\left (n -1\right ) \ln \left (y \right )}{n}+c_{1}}\\ &=c_{1} {\mathrm e}^{\frac {\left (n -1\right ) \ln \left (y \right )}{n}} \end {align*}
Which simplifies to \[ p \left (y \right ) = c_{1} y \,y^{-\frac {1}{n}} \] 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 } = c_{1} y \,y^{-\frac {1}{n}} \end {align*}
Integrating both sides gives \begin{align*} \int \frac {y^{\frac {1}{n}}}{c_{1} y}d y &= \int d x \\ \frac {n y^{\frac {1}{n}}}{c_{1}}&=x +c_{2} \\ \end{align*}
The solution(s) found are the following \begin{align*} \tag{1} y &= {\mathrm e}^{n \ln \left (\frac {c_{1} \left (x +c_{2} \right )}{n}\right )} \\ \end{align*}
Verification of solutions
\[ y = {\mathrm e}^{n \ln \left (\frac {c_{1} \left (x +c_{2} \right )}{n}\right )} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & n y \left (\frac {d}{d x}y^{\prime }\right )+\left (-n y^{\prime }+y^{\prime }\right ) y^{\prime }=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 2 \\ {} & {} & \frac {d}{d x}y^{\prime } \\ \bullet & {} & \textrm {Define new dependent variable}\hspace {3pt} u \\ {} & {} & u \left (x \right )=y^{\prime } \\ \bullet & {} & \textrm {Compute}\hspace {3pt} \frac {d}{d x}y^{\prime } \\ {} & {} & u^{\prime }\left (x \right )=y^{\prime \prime } \\ \bullet & {} & \textrm {Use chain rule on the lhs}\hspace {3pt} \\ {} & {} & y^{\prime } \left (\frac {d}{d y}u \left (y \right )\right )=y^{\prime \prime } \\ \bullet & {} & \textrm {Substitute in the definition of}\hspace {3pt} u \\ {} & {} & u \left (y \right ) \left (\frac {d}{d y}u \left (y \right )\right )=y^{\prime \prime } \\ \bullet & {} & \textrm {Make substitutions}\hspace {3pt} y^{\prime }=u \left (y \right ),\frac {d}{d x}y^{\prime }=u \left (y \right ) \left (\frac {d}{d y}u \left (y \right )\right )\hspace {3pt}\textrm {to reduce order of ODE}\hspace {3pt} \\ {} & {} & n y u \left (y \right ) \left (\frac {d}{d y}u \left (y \right )\right )+\left (-n u \left (y \right )+u \left (y \right )\right ) u \left (y \right )=0 \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d y}u \left (y \right )=-\frac {-n u \left (y \right )+u \left (y \right )}{n y} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {\frac {d}{d y}u \left (y \right )}{-n u \left (y \right )+u \left (y \right )}=-\frac {1}{y n} \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} y \\ {} & {} & \int \frac {\frac {d}{d y}u \left (y \right )}{-n u \left (y \right )+u \left (y \right )}d y =\int -\frac {1}{y n}d y +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \frac {\ln \left (u \left (y \right )\right )}{-n +1}=-\frac {\ln \left (y \right )}{n}+c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} u \left (y \right ) \\ {} & {} & u \left (y \right )={\mathrm e}^{\frac {-c_{1} n^{2}+\ln \left (y \right ) n +c_{1} n -\ln \left (y \right )}{n}} \\ \bullet & {} & \textrm {Solve 1st ODE for}\hspace {3pt} u \left (y \right ) \\ {} & {} & u \left (y \right )={\mathrm e}^{\frac {-c_{1} n^{2}+\ln \left (y \right ) n +c_{1} n -\ln \left (y \right )}{n}} \\ \bullet & {} & \textrm {Revert to original variables with substitution}\hspace {3pt} u \left (y \right )=y^{\prime },y =y \\ {} & {} & y^{\prime }={\mathrm e}^{\frac {-c_{1} n^{2}+\ln \left (y\right ) n +c_{1} n -\ln \left (y\right )}{n}} \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }={\mathrm e}^{\frac {-c_{1} n^{2}+\ln \left (y\right ) n +c_{1} n -\ln \left (y\right )}{n}} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{{\mathrm e}^{\frac {-c_{1} n^{2}+\ln \left (y\right ) n +c_{1} n -\ln \left (y\right )}{n}}}=1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{{\mathrm e}^{\frac {-c_{1} n^{2}+\ln \left (y\right ) n +c_{1} n -\ln \left (y\right )}{n}}}d x =\int 1d x +c_{2} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \frac {{\mathrm e}^{c_{1} \left (n -1\right )} n}{y^{-\frac {1}{n}}}=x +c_{2} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y={\mathrm e}^{-\left (c_{1} n +\ln \left (\frac {n}{x +c_{2}}\right )-c_{1} \right ) n} \end {array} \]
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: 19
dsolve(n*y(x)*diff(diff(y(x),x),x)-(n-1)*diff(y(x),x)^2=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= 0 \\ y \left (x \right ) &= \left (\frac {c_{1} x +c_{2}}{n}\right )^{n} \\ \end{align*}
✓ Solution by Mathematica
Time used: 1.022 (sec). Leaf size: 17
DSolve[(1 - n)*y'[x]^2 + n*y[x]*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to c_2 (x-c_1 n){}^n \]