Internal problem ID [9977]
Internal file name [OUTPUT/8924_Monday_June_06_2022_05_52_58_AM_3804322/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1656 (book 6.65).
ODE order: 2.
ODE degree: 2.
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]]
\[ \boxed {y^{\prime \prime }-a y \left ({y^{\prime }}^{2}+1\right )^{\frac {3}{2}}=0} \]
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*} p \left (y \right ) \left (\frac {d}{d y}p \left (y \right )\right )-a y \left (p \left (y \right )^{2}+1\right )^{\frac {3}{2}} = 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 {a y \left (p^{2}+1\right )^{\frac {3}{2}}}{p} \end {align*}
Where \(f(y)=y a\) and \(g(p)=\frac {\left (p^{2}+1\right )^{\frac {3}{2}}}{p}\). Integrating both sides gives \begin{align*} \frac {1}{\frac {\left (p^{2}+1\right )^{\frac {3}{2}}}{p}} \,dp &= y a \,d y \\ \int { \frac {1}{\frac {\left (p^{2}+1\right )^{\frac {3}{2}}}{p}} \,dp} &= \int {y a \,d y} \\ -\frac {1}{\sqrt {p^{2}+1}}&=\frac {a \,y^{2}}{2}+c_{1} \\ \end{align*} The solution is \[ -\frac {1}{\sqrt {p \left (y \right )^{2}+1}}-\frac {a \,y^{2}}{2}-c_{1} = 0 \] For solution (1) found earlier, since \(p=y^{\prime }\) then we now have a new first order ode to solve which is \begin {align*} -\frac {1}{\sqrt {{y^{\prime }}^{2}+1}}-\frac {a y^{2}}{2}-c_{1} = 0 \end {align*}
Solving the given ode for \(y^{\prime }\) results in \(2\) differential equations to solve. Each one of these will generate a solution. The equations generated are \begin {align*} y^{\prime }&=\frac {\sqrt {-a^{2} y^{4}-4 y^{2} a c_{1} -4 c_{1}^{2}+4}}{a y^{2}+2 c_{1}} \tag {1} \\ y^{\prime }&=-\frac {\sqrt {-a^{2} y^{4}-4 y^{2} a c_{1} -4 c_{1}^{2}+4}}{a y^{2}+2 c_{1}} \tag {2} \end {align*}
Now each one of the above ODE is solved.
Solving equation (1)
Integrating both sides gives \begin {align*} \int \frac {a \,y^{2}+2 c_{1}}{\sqrt {-a^{2} y^{4}-4 a c_{1} y^{2}-4 c_{1}^{2}+4}}d y &= \int {dx}\\ \int _{}^{y}\frac {\textit {\_a}^{2} a +2 c_{1}}{\sqrt {-\textit {\_a}^{4} a^{2}-4 \textit {\_a}^{2} a c_{1} -4 c_{1}^{2}+4}}d \textit {\_a}&= x +c_{2} \end {align*}
Solving equation (2)
Integrating both sides gives \begin {align*} \int -\frac {a \,y^{2}+2 c_{1}}{\sqrt {-a^{2} y^{4}-4 a c_{1} y^{2}-4 c_{1}^{2}+4}}d y &= \int {dx}\\ \int _{}^{y}-\frac {\textit {\_a}^{2} a +2 c_{1}}{\sqrt {-\textit {\_a}^{4} a^{2}-4 \textit {\_a}^{2} a c_{1} -4 c_{1}^{2}+4}}d \textit {\_a}&= c_{3} +x \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} \int _{}^{y}\frac {\textit {\_a}^{2} a +2 c_{1}}{\sqrt {-\textit {\_a}^{4} a^{2}-4 \textit {\_a}^{2} a c_{1} -4 c_{1}^{2}+4}}d \textit {\_a} &= x +c_{2} \\ \tag{2} \int _{}^{y}-\frac {\textit {\_a}^{2} a +2 c_{1}}{\sqrt {-\textit {\_a}^{4} a^{2}-4 \textit {\_a}^{2} a c_{1} -4 c_{1}^{2}+4}}d \textit {\_a} &= c_{3} +x \\ \end{align*}
Verification of solutions
\[ \int _{}^{y}\frac {\textit {\_a}^{2} a +2 c_{1}}{\sqrt {-\textit {\_a}^{4} a^{2}-4 \textit {\_a}^{2} a c_{1} -4 c_{1}^{2}+4}}d \textit {\_a} = x +c_{2} \] Verified OK.
\[ \int _{}^{y}-\frac {\textit {\_a}^{2} a +2 c_{1}}{\sqrt {-\textit {\_a}^{4} a^{2}-4 \textit {\_a}^{2} a c_{1} -4 c_{1}^{2}+4}}d \textit {\_a} = c_{3} +x \] 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 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)-a*_a*(_b(_a)^2+1)^(3/2) = 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: 0.109 (sec). Leaf size: 104
dsolve(diff(diff(y(x),x),x)-a*y(x)*(diff(y(x),x)^2+1)^(3/2)=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= -i x +c_{1} \\ y \left (x \right ) &= i x +c_{1} \\ a \left (\int _{}^{y \left (x \right )}\frac {\textit {\_a}^{2}+2 c_{1}}{\sqrt {4-\left (\textit {\_a}^{2}+2 c_{1} \right )^{2} a^{2}}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ -a \left (\int _{}^{y \left (x \right )}\frac {\textit {\_a}^{2}+2 c_{1}}{\sqrt {4-\left (\textit {\_a}^{2}+2 c_{1} \right )^{2} a^{2}}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 3.326 (sec). Leaf size: 1104
DSolve[-(a*y[x]*(1 + y'[x]^2)^(3/2)) + y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \text {InverseFunction}\left [-\frac {\sqrt {\frac {\text {$\#$1}^2 a-2+2 c_1}{-1+c_1}} \sqrt {\frac {\text {$\#$1}^2 a+2+2 c_1}{1+c_1}} \left (\operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {a}{2 c_1+2}} \text {$\#$1}\right ),\frac {c_1+1}{c_1-1}\right )+(-1+c_1) E\left (i \text {arcsinh}\left (\sqrt {\frac {a}{2 c_1+2}} \text {$\#$1}\right )|\frac {c_1+1}{c_1-1}\right )\right )}{\sqrt {\frac {a}{2+2 c_1}} \sqrt {\text {$\#$1}^4 a^2+4 \text {$\#$1}^2 a c_1-4+4 c_1{}^2}}\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\frac {\sqrt {\frac {\text {$\#$1}^2 a-2+2 c_1}{-1+c_1}} \sqrt {\frac {\text {$\#$1}^2 a+2+2 c_1}{1+c_1}} \left (\operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {a}{2 c_1+2}} \text {$\#$1}\right ),\frac {c_1+1}{c_1-1}\right )+(-1+c_1) E\left (i \text {arcsinh}\left (\sqrt {\frac {a}{2 c_1+2}} \text {$\#$1}\right )|\frac {c_1+1}{c_1-1}\right )\right )}{\sqrt {\frac {a}{2+2 c_1}} \sqrt {\text {$\#$1}^4 a^2+4 \text {$\#$1}^2 a c_1-4+4 c_1{}^2}}\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [-\frac {\sqrt {\frac {\text {$\#$1}^2 a-2+2 (-1) c_1}{-1-c_1}} \sqrt {\frac {\text {$\#$1}^2 a+2+2 (-1) c_1}{1-c_1}} \left (\operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {a}{2 (-1) c_1+2}} \text {$\#$1}\right ),\frac {1-c_1}{-c_1-1}\right )+(-1-c_1) E\left (i \text {arcsinh}\left (\sqrt {\frac {a}{2 (-1) c_1+2}} \text {$\#$1}\right )|\frac {1-c_1}{-c_1-1}\right )\right )}{\sqrt {\frac {a}{2+2 (-1) c_1}} \sqrt {\text {$\#$1}^4 a^2+4 \text {$\#$1}^2 a (-c_1)-4+4 (-c_1){}^2}}\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\frac {\sqrt {\frac {\text {$\#$1}^2 a-2+2 (-1) c_1}{-1-c_1}} \sqrt {\frac {\text {$\#$1}^2 a+2+2 (-1) c_1}{1-c_1}} \left (\operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {a}{2 (-1) c_1+2}} \text {$\#$1}\right ),\frac {1-c_1}{-c_1-1}\right )+(-1-c_1) E\left (i \text {arcsinh}\left (\sqrt {\frac {a}{2 (-1) c_1+2}} \text {$\#$1}\right )|\frac {1-c_1}{-c_1-1}\right )\right )}{\sqrt {\frac {a}{2+2 (-1) c_1}} \sqrt {\text {$\#$1}^4 a^2+4 \text {$\#$1}^2 a (-c_1)-4+4 (-c_1){}^2}}\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [-\frac {\sqrt {\frac {\text {$\#$1}^2 a-2+2 c_1}{-1+c_1}} \sqrt {\frac {\text {$\#$1}^2 a+2+2 c_1}{1+c_1}} \left (\operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {a}{2 c_1+2}} \text {$\#$1}\right ),\frac {c_1+1}{c_1-1}\right )+(-1+c_1) E\left (i \text {arcsinh}\left (\sqrt {\frac {a}{2 c_1+2}} \text {$\#$1}\right )|\frac {c_1+1}{c_1-1}\right )\right )}{\sqrt {\frac {a}{2+2 c_1}} \sqrt {\text {$\#$1}^4 a^2+4 \text {$\#$1}^2 a c_1-4+4 c_1{}^2}}\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\frac {\sqrt {\frac {\text {$\#$1}^2 a-2+2 c_1}{-1+c_1}} \sqrt {\frac {\text {$\#$1}^2 a+2+2 c_1}{1+c_1}} \left (\operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {a}{2 c_1+2}} \text {$\#$1}\right ),\frac {c_1+1}{c_1-1}\right )+(-1+c_1) E\left (i \text {arcsinh}\left (\sqrt {\frac {a}{2 c_1+2}} \text {$\#$1}\right )|\frac {c_1+1}{c_1-1}\right )\right )}{\sqrt {\frac {a}{2+2 c_1}} \sqrt {\text {$\#$1}^4 a^2+4 \text {$\#$1}^2 a c_1-4+4 c_1{}^2}}\&\right ][x+c_2] \\ \end{align*}