Internal problem ID [8959]
Internal file name [OUTPUT/7894_Monday_June_06_2022_12_51_42_AM_45938527/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 625.
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.
\[ \boxed {y^{\prime }-\frac {i x^{2} \left (i-2 \sqrt {-x^{3}+6 y}\right )}{2}=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \mathrm {I} x^{2}-2 x^{2} \sqrt {-x^{3}+6 y}+2 \,\mathrm {I} y^{\prime }=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-\frac {\mathrm {I}}{2} \left (-\mathrm {I} x^{2}+2 x^{2} \sqrt {-x^{3}+6 y}\right ) \end {array} \]
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 = 5`[0, -1/2*I+1/2*(-x^3+6*y)^(1/2)]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 55
dsolve(diff(y(x),x) = 1/2*I*x^2*(I-2*(-x^3+6*y(x))^(1/2)),y(x), singsol=all)
\[ i \ln \left (x^{3}-6 y \left (x \right )-1\right )+2 \sqrt {-x^{3}+6 y \left (x \right )}-2 \arctan \left (\sqrt {-x^{3}+6 y \left (x \right )}\right )+2 i x^{3}-c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 11.298 (sec). Leaf size: 69
DSolve[y'[x] == (I/2)*x^2*(I - 2*Sqrt[-x^3 + 6*y[x]]),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{6} \left (-W\left (-i e^{-x^3-1-6 c_1}\right ){}^2-2 W\left (-i e^{-x^3-1-6 c_1}\right )+x^3-1\right ) \\ y(x)\to \frac {1}{6} \left (x^3-1\right ) \\ \end{align*}