Internal problem ID [8948]
Internal file name [OUTPUT/7883_Monday_June_06_2022_12_50_16_AM_86998591/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 614.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[[_1st_order, `_with_symmetry_[F(x),G(y)]`]]
Unable to solve or complete the solution.
\[ \boxed {y^{\prime }-\frac {x \left (-1+a \right ) \left (a +1\right )}{y+F \left (\frac {y^{2}}{2}-\frac {a^{2} x^{2}}{2}+\frac {x^{2}}{2}\right ) a^{2}-F \left (\frac {y^{2}}{2}-\frac {a^{2} x^{2}}{2}+\frac {x^{2}}{2}\right )}=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime } F \left (\frac {y^{2}}{2}-\frac {a^{2} x^{2}}{2}+\frac {x^{2}}{2}\right ) a^{2}-a^{2} x +y y^{\prime }-y^{\prime } F \left (\frac {y^{2}}{2}-\frac {a^{2} x^{2}}{2}+\frac {x^{2}}{2}\right )+x =0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {a^{2} x -x}{y+F \left (\frac {y^{2}}{2}-\frac {a^{2} x^{2}}{2}+\frac {x^{2}}{2}\right ) a^{2}-F \left (\frac {y^{2}}{2}-\frac {a^{2} x^{2}}{2}+\frac {x^{2}}{2}\right )} \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying homogeneous types: differential order: 1; looking for linear symmetries trying exact Looking for potential symmetries trying an equivalence to an Abel ODE trying 1st order ODE linearizable_by_differentiation --- Trying Lie symmetry methods, 1st order --- `, `-> Computing symmetries using: way = 3 `, `-> Computing symmetries using: way = 4`[y/x, a^2-1]
✓ Solution by Maple
Time used: 0.094 (sec). Leaf size: 93
dsolve(diff(y(x),x) = x*(a-1)*(a+1)/(y(x)+F(1/2*y(x)^2-1/2*a^2*x^2+1/2*x^2)*a^2-F(1/2*y(x)^2-1/2*a^2*x^2+1/2*x^2)),y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \operatorname {RootOf}\left (F \left (\frac {1}{2} \textit {\_Z}^{2}-\frac {1}{2} a^{2} x^{2}+\frac {1}{2} x^{2}\right )\right ) \\ \frac {\int _{}^{-a^{2} x^{2}+x^{2}+y \left (x \right )^{2}}\frac {1}{F \left (\frac {\textit {\_a}}{2}\right )}d \textit {\_a} +\left (2 a^{2}-2\right ) y \left (x \right )-2 c_{1} a^{4}+4 a^{2} c_{1} -2 c_{1}}{2 a^{4}-4 a^{2}+2} &= 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.323 (sec). Leaf size: 177
DSolve[y'[x] == ((-1 + a)*(1 + a)*x)/(-F[x^2/2 - (a^2*x^2)/2 + y[x]^2/2] + a^2*F[x^2/2 - (a^2*x^2)/2 + y[x]^2/2] + y[x]),y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\int _1^{y(x)}\left (\frac {K[2]}{(a-1) (a+1) F\left (-\frac {1}{2} a^2 x^2+\frac {x^2}{2}+\frac {K[2]^2}{2}\right )}-\int _1^x\frac {K[1] K[2] F'\left (-\frac {1}{2} a^2 K[1]^2+\frac {K[1]^2}{2}+\frac {K[2]^2}{2}\right )}{F\left (-\frac {1}{2} a^2 K[1]^2+\frac {K[1]^2}{2}+\frac {K[2]^2}{2}\right )^2}dK[1]+1\right )dK[2]+\int _1^x-\frac {K[1]}{F\left (-\frac {1}{2} a^2 K[1]^2+\frac {K[1]^2}{2}+\frac {y(x)^2}{2}\right )}dK[1]=c_1,y(x)\right ] \]