Internal problem ID [9004]
Internal file name [OUTPUT/7939_Monday_June_06_2022_12_58_25_AM_49317327/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 670.
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.
\[ \boxed {y^{\prime }-\frac {i x \left (i-2 \sqrt {-x^{2}+4 \ln \left (a \right )+4 \ln \left (y\right )}\right ) y}{2}=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \mathrm {I} y x -2 x y \sqrt {-x^{2}+4 \ln \left (a \right )+4 \ln \left (y\right )}+2 \,\mathrm {I} y^{\prime }=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-\frac {\mathrm {I}}{2} \left (-\mathrm {I} y x +2 x y \sqrt {-x^{2}+4 \ln \left (a \right )+4 \ln \left (y\right )}\right ) \end {array} \]
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*(-x^2+4*ln(a)+4*ln(y))^(1/2)*y+1/2*y]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 72
dsolve(diff(y(x),x) = 1/2*I*x*(I-2*(-x^2+4*ln(a)+4*ln(y(x)))^(1/2))*y(x),y(x), singsol=all)
\[ -\frac {i \ln \left (-x^{2}+4 \ln \left (a \right )+4 \ln \left (y \left (x \right )\right )+1\right )}{4}-\frac {\sqrt {-x^{2}+4 \ln \left (a \right )+4 \ln \left (y \left (x \right )\right )}}{2}+\frac {\arctan \left (\sqrt {-x^{2}+4 \ln \left (a \right )+4 \ln \left (y \left (x \right )\right )}\right )}{2}-\frac {i x^{2}}{2}-c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 9.83 (sec). Leaf size: 86
DSolve[y'[x] == (I/2)*x*(I - 2*Sqrt[-x^2 + 4*Log[a] + 4*Log[y[x]]])*y[x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \exp \left (\frac {1}{4} \left (-4 \log (a)-W\left (i e^{-x^2-1-4 c_1}\right ){}^2-2 W\left (i e^{-x^2-1-4 c_1}\right )+x^2-1\right )\right ) \\ y(x)\to 0 \\ y(x)\to \frac {e^{\frac {1}{4} \left (x^2-1\right )}}{a} \\ \end{align*}