Internal problem ID [6849]
Internal file name [OUTPUT/6096_Thursday_July_28_2022_04_30_22_AM_21244975/index.tex]
Book: Elementary differential equations. By Earl D. Rainville, Phillip E. Bedient. Macmilliam
Publishing Co. NY. 6th edition. 1981.
Section: CHAPTER 16. Nonlinear equations. Section 101. Independent variable missing.
EXERCISES Page 324
Problem number: 32.
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], [_2nd_order, _reducible, _mu_x_y1]]
\[ \boxed {\left (1+y^{2}\right ) y^{\prime \prime }+{y^{\prime }}^{3}+y^{\prime }=0} \]
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*} \left (y^{2}+1\right ) p \left (y \right ) \left (\frac {d}{d y}p \left (y \right )\right )+p \left (y \right ) \left (1+p \left (y \right )^{2}\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^{2}-1}{y^{2}+1} \end {align*}
Where \(f(y)=\frac {1}{y^{2}+1}\) and \(g(p)=-p^{2}-1\). Integrating both sides gives \begin{align*} \frac {1}{-p^{2}-1} \,dp &= \frac {1}{y^{2}+1} \,d y \\ \int { \frac {1}{-p^{2}-1} \,dp} &= \int {\frac {1}{y^{2}+1} \,d y} \\ -\arctan \left (p \right )&=\arctan \left (y \right )+c_{1} \\ \end{align*} The solution is \[ -\arctan \left (p \left (y \right )\right )-\arctan \left (y \right )-c_{1} = 0 \] For solution (1) found earlier, since \(p=y^{\prime }\) then we now have a new first order ode to solve which is \begin {align*} -\arctan \left (y^{\prime }\right )-\arctan \left (y\right )-c_{1} = 0 \end {align*}
Integrating both sides gives \begin {align*} \int -\frac {1}{\tan \left (\arctan \left (y \right )+c_{1} \right )}d y &= \int {dx}\\ \int _{}^{y}-\frac {1}{\tan \left (\arctan \left (\textit {\_a} \right )+c_{1} \right )}d \textit {\_a}&= x +c_{2} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} \int _{}^{y}-\frac {1}{\tan \left (\arctan \left (\textit {\_a} \right )+c_{1} \right )}d \textit {\_a} &= x +c_{2} \\ \end{align*}
Verification of solutions
\[ \int _{}^{y}-\frac {1}{\tan \left (\arctan \left (\textit {\_a} \right )+c_{1} \right )}d \textit {\_a} = x +c_{2} \] Verified OK.
Maple trace
`Methods for second order ODEs: --- Trying classification methods --- trying 2nd order Liouville trying 2nd order WeierstrassP trying 2nd order JacobiSN differential order: 2; trying a linearization to 3rd order --- trying a change of variables {x -> y(x), y(x) -> x} differential order: 2; trying a linearization to 3rd order trying 2nd order ODE linearizable_by_differentiation trying 2nd order, 2 integrating factors of the form mu(x,y) trying differential order: 2; missing variables `, `-> Computing symmetries using: way = 3 `, `-> Computing symmetries using: way = exp_sym -> Calling odsolve with the ODE`, (diff(_b(_a), _a))*_b(_a)+_b(_a)*(1+_b(_a)^2)/(_a^2+1) = 0, _b(_a)` *** Sublevel 2 *** Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear trying Bernoulli trying separable <- separable successful <- differential order: 2; canonical coordinates successful <- differential order 2; missing variables successful`
✓ Solution by Maple
Time used: 0.187 (sec). Leaf size: 118
dsolve((1+y(x)^2)*diff(y(x),x$2)+diff(y(x),x)^3+diff(y(x),x)=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= -i \\ y \left (x \right ) &= i \\ y \left (x \right ) &= c_{1} \\ y \left (x \right ) &= \frac {i c_{1} -i-{\mathrm e}^{\frac {-4 \operatorname {LambertW}\left (-\frac {i {\mathrm e}^{\frac {\left (-c_{2} -x +1\right ) c_{1}^{2}+\left (-2 c_{2} -2 x -2\right ) c_{1} -x -c_{2} +1}{4 c_{1}}} \left (c_{1} -1\right )}{4 c_{1}}\right ) c_{1} +\left (-c_{2} -x +1\right ) c_{1}^{2}+\left (-2 c_{2} -2 x -2\right ) c_{1} -x -c_{2} +1}{4 c_{1}}}}{c_{1} +1} \\ \end{align*}
✓ Solution by Mathematica
Time used: 57.998 (sec). Leaf size: 56
DSolve[(1+y[x]^2)*y''[x]+(y'[x])^3+y'[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \csc (c_1) \sec (c_1) W\left (\sin (c_1) e^{-\left ((x+c_2) \cos ^2(c_1)\right )-\sin ^2(c_1)}\right )+\tan (c_1) \\ y(x)\to e^{-x-c_2} \\ \end{align*}