Internal problem ID [9976]
Internal file name [OUTPUT/8923_Monday_June_06_2022_05_52_39_AM_63686247/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1655 (book 6.64).
ODE order: 2.
ODE degree: 2.
The type(s) of ODE detected by this program : "second_order_ode_missing_y"
Maple gives the following as the ode type
[[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_y_y1]]
\[ \boxed {y^{\prime \prime }-2 a x \left ({y^{\prime }}^{2}+1\right )^{\frac {3}{2}}=0} \]
This is second order ode with missing dependent variable \(y\). Let \begin {align*} p(x) &= y^{\prime } \end {align*}
Then \begin {align*} p'(x) &= y^{\prime \prime } \end {align*}
Hence the ode becomes \begin {align*} p^{\prime }\left (x \right )-2 a x \left (p \left (x \right )^{2}+1\right )^{\frac {3}{2}} = 0 \end {align*}
Which is now solve for \(p(x)\) as first order ode. In canonical form the ODE is \begin {align*} p' &= F(x,p)\\ &= f( x) g(p)\\ &= 2 a x \left (p^{2}+1\right )^{\frac {3}{2}} \end {align*}
Where \(f(x)=2 x a\) and \(g(p)=\left (p^{2}+1\right )^{\frac {3}{2}}\). Integrating both sides gives \begin{align*} \frac {1}{\left (p^{2}+1\right )^{\frac {3}{2}}} \,dp &= 2 x a \,d x \\ \int { \frac {1}{\left (p^{2}+1\right )^{\frac {3}{2}}} \,dp} &= \int {2 x a \,d x} \\ \frac {p}{\sqrt {p^{2}+1}}&=x^{2} a +c_{1} \\ \end{align*} The solution is \[ \frac {p \left (x \right )}{\sqrt {p \left (x \right )^{2}+1}}-x^{2} a -c_{1} = 0 \] 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 {y^{\prime }}{\sqrt {{y^{\prime }}^{2}+1}}-x^{2} a -c_{1} = 0 \end {align*}
Integrating both sides gives \begin {align*} y &= \int { x^{2} a \sqrt {-\frac {1}{a^{2} x^{4}+2 a c_{1} x^{2}+c_{1}^{2}-1}}+c_{1} \sqrt {-\frac {1}{a^{2} x^{4}+2 a c_{1} x^{2}+c_{1}^{2}-1}}\,\mathop {\mathrm {d}x}}\\ &= \frac {c_{1} \sqrt {-\frac {1}{a^{2} x^{4}+2 a c_{1} x^{2}+c_{1}^{2}-1}}\, \sqrt {\frac {x^{2} a +c_{1} +1}{c_{1} +1}}\, \sqrt {\frac {x^{2} a +c_{1} -1}{c_{1} -1}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {a}{c_{1} +1}}, \sqrt {\frac {c_{1} +1}{c_{1} -1}}\right )}{\sqrt {-\frac {a}{c_{1} +1}}}+\frac {\left (-\operatorname {EllipticF}\left (x \sqrt {-\frac {a}{c_{1} +1}}, \sqrt {\frac {c_{1} +1}{c_{1} -1}}\right )+\operatorname {EllipticE}\left (x \sqrt {-\frac {a}{c_{1} +1}}, \sqrt {\frac {c_{1} +1}{c_{1} -1}}\right )\right ) \sqrt {\frac {x^{2} a +c_{1} -1}{c_{1} -1}}\, \sqrt {\frac {x^{2} a +c_{1} +1}{c_{1} +1}}\, \left (c_{1} -1\right ) \sqrt {-\frac {1}{a^{2} x^{4}+2 a c_{1} x^{2}+c_{1}^{2}-1}}}{\sqrt {-\frac {a}{c_{1} +1}}}+c_{2} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {c_{1} \sqrt {-\frac {1}{a^{2} x^{4}+2 a c_{1} x^{2}+c_{1}^{2}-1}}\, \sqrt {\frac {x^{2} a +c_{1} +1}{c_{1} +1}}\, \sqrt {\frac {x^{2} a +c_{1} -1}{c_{1} -1}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {a}{c_{1} +1}}, \sqrt {\frac {c_{1} +1}{c_{1} -1}}\right )}{\sqrt {-\frac {a}{c_{1} +1}}}+\frac {\left (-\operatorname {EllipticF}\left (x \sqrt {-\frac {a}{c_{1} +1}}, \sqrt {\frac {c_{1} +1}{c_{1} -1}}\right )+\operatorname {EllipticE}\left (x \sqrt {-\frac {a}{c_{1} +1}}, \sqrt {\frac {c_{1} +1}{c_{1} -1}}\right )\right ) \sqrt {\frac {x^{2} a +c_{1} -1}{c_{1} -1}}\, \sqrt {\frac {x^{2} a +c_{1} +1}{c_{1} +1}}\, \left (c_{1} -1\right ) \sqrt {-\frac {1}{a^{2} x^{4}+2 a c_{1} x^{2}+c_{1}^{2}-1}}}{\sqrt {-\frac {a}{c_{1} +1}}}+c_{2} \\ \end{align*}
Verification of solutions
\[ y = \frac {c_{1} \sqrt {-\frac {1}{a^{2} x^{4}+2 a c_{1} x^{2}+c_{1}^{2}-1}}\, \sqrt {\frac {x^{2} a +c_{1} +1}{c_{1} +1}}\, \sqrt {\frac {x^{2} a +c_{1} -1}{c_{1} -1}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {a}{c_{1} +1}}, \sqrt {\frac {c_{1} +1}{c_{1} -1}}\right )}{\sqrt {-\frac {a}{c_{1} +1}}}+\frac {\left (-\operatorname {EllipticF}\left (x \sqrt {-\frac {a}{c_{1} +1}}, \sqrt {\frac {c_{1} +1}{c_{1} -1}}\right )+\operatorname {EllipticE}\left (x \sqrt {-\frac {a}{c_{1} +1}}, \sqrt {\frac {c_{1} +1}{c_{1} -1}}\right )\right ) \sqrt {\frac {x^{2} a +c_{1} -1}{c_{1} -1}}\, \sqrt {\frac {x^{2} a +c_{1} +1}{c_{1} +1}}\, \left (c_{1} -1\right ) \sqrt {-\frac {1}{a^{2} x^{4}+2 a c_{1} x^{2}+c_{1}^{2}-1}}}{\sqrt {-\frac {a}{c_{1} +1}}}+c_{2} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y^{\prime }=2 a x \left ({y^{\prime }}^{2}+1\right )^{\frac {3}{2}} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 2 \\ {} & {} & \frac {d}{d x}y^{\prime } \\ \bullet & {} & \textrm {Make substitution}\hspace {3pt} u =y^{\prime }\hspace {3pt}\textrm {to reduce order of ODE}\hspace {3pt} \\ {} & {} & u^{\prime }\left (x \right )=2 a x \left (u \left (x \right )^{2}+1\right )^{\frac {3}{2}} \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & u^{\prime }\left (x \right )=2 a x \left (u \left (x \right )^{2}+1\right )^{\frac {3}{2}} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {u^{\prime }\left (x \right )}{\left (u \left (x \right )^{2}+1\right )^{\frac {3}{2}}}=2 x a \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {u^{\prime }\left (x \right )}{\left (u \left (x \right )^{2}+1\right )^{\frac {3}{2}}}d x =\int 2 x a d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \frac {u \left (x \right )}{\sqrt {u \left (x \right )^{2}+1}}=x^{2} a +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} u \left (x \right ) \\ {} & {} & u \left (x \right )=x^{2} a \sqrt {-\frac {1}{a^{2} x^{4}+2 c_{1} a \,x^{2}+c_{1}^{2}-1}}+c_{1} \sqrt {-\frac {1}{a^{2} x^{4}+2 c_{1} a \,x^{2}+c_{1}^{2}-1}} \\ \bullet & {} & \textrm {Solve 1st ODE for}\hspace {3pt} u \left (x \right ) \\ {} & {} & u \left (x \right )=x^{2} a \sqrt {-\frac {1}{a^{2} x^{4}+2 c_{1} a \,x^{2}+c_{1}^{2}-1}}+c_{1} \sqrt {-\frac {1}{a^{2} x^{4}+2 c_{1} a \,x^{2}+c_{1}^{2}-1}} \\ \bullet & {} & \textrm {Make substitution}\hspace {3pt} u =y^{\prime } \\ {} & {} & y^{\prime }=x^{2} a \sqrt {-\frac {1}{a^{2} x^{4}+2 c_{1} a \,x^{2}+c_{1}^{2}-1}}+c_{1} \sqrt {-\frac {1}{a^{2} x^{4}+2 c_{1} a \,x^{2}+c_{1}^{2}-1}} \\ \bullet & {} & \textrm {Integrate both sides to solve for}\hspace {3pt} y \\ {} & {} & \int y^{\prime }d x =\int \left (x^{2} a \sqrt {-\frac {1}{a^{2} x^{4}+2 c_{1} a \,x^{2}+c_{1}^{2}-1}}+c_{1} \sqrt {-\frac {1}{a^{2} x^{4}+2 c_{1} a \,x^{2}+c_{1}^{2}-1}}\right )d x +c_{2} \\ \bullet & {} & \textrm {Compute integrals}\hspace {3pt} \\ {} & {} & y=\frac {c_{1} \sqrt {-\frac {1}{a^{2} x^{4}+2 c_{1} a \,x^{2}+c_{1}^{2}-1}}\, \sqrt {\frac {x^{2} a +c_{1} +1}{c_{1} +1}}\, \sqrt {\frac {x^{2} a +c_{1} -1}{c_{1} -1}}\, \mathit {EllipticF}\left (x \sqrt {-\frac {a}{c_{1} +1}}, \sqrt {\frac {c_{1} +1}{c_{1} -1}}\right )}{\sqrt {-\frac {a}{c_{1} +1}}}+\frac {\left (-\mathit {EllipticF}\left (x \sqrt {-\frac {a}{c_{1} +1}}, \sqrt {\frac {c_{1} +1}{c_{1} -1}}\right )+\mathit {EllipticE}\left (x \sqrt {-\frac {a}{c_{1} +1}}, \sqrt {\frac {c_{1} +1}{c_{1} -1}}\right )\right ) \sqrt {\frac {x^{2} a +c_{1} -1}{c_{1} -1}}\, \sqrt {\frac {x^{2} a +c_{1} +1}{c_{1} +1}}\, \left (c_{1} -1\right ) \sqrt {-\frac {1}{a^{2} x^{4}+2 c_{1} a \,x^{2}+c_{1}^{2}-1}}}{\sqrt {-\frac {a}{c_{1} +1}}}+c_{2} \end {array} \]
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) = 2*a*_a*(_b(_a)^2+1)^(3/2), _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: 57
dsolve(diff(diff(y(x),x),x)-2*a*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} \\ y \left (x \right ) &= a \left (\int \sqrt {-\frac {1}{-1+\left (x^{2}+2 c_{1} \right )^{2} a^{2}}}\, \left (x^{2}+2 c_{1} \right )d x \right )+c_{2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 60.462 (sec). Leaf size: 308
DSolve[-2*a*x*(1 + y'[x]^2)^(3/2) + y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_2-\frac {\sqrt {\frac {a x^2-1+c_1}{-1+c_1}} \sqrt {\frac {a x^2+1+c_1}{1+c_1}} \left (\operatorname {EllipticF}\left (i \text {arcsinh}\left (x \sqrt {\frac {a}{c_1+1}}\right ),\frac {c_1+1}{c_1-1}\right )+(-1+c_1) E\left (i \text {arcsinh}\left (x \sqrt {\frac {a}{c_1+1}}\right )|\frac {c_1+1}{c_1-1}\right )\right )}{\sqrt {\frac {a}{1+c_1}} \sqrt {a^2 x^4+2 a c_1 x^2-1+c_1{}^2}} \\ y(x)\to \frac {\sqrt {\frac {a x^2-1+c_1}{-1+c_1}} \sqrt {\frac {a x^2+1+c_1}{1+c_1}} \left (\operatorname {EllipticF}\left (i \text {arcsinh}\left (x \sqrt {\frac {a}{c_1+1}}\right ),\frac {c_1+1}{c_1-1}\right )+(-1+c_1) E\left (i \text {arcsinh}\left (x \sqrt {\frac {a}{c_1+1}}\right )|\frac {c_1+1}{c_1-1}\right )\right )}{\sqrt {\frac {a}{1+c_1}} \sqrt {a^2 x^4+2 a c_1 x^2-1+c_1{}^2}}+c_2 \\ \end{align*}