Internal problem ID [8997]
Internal file name [OUTPUT/7932_Monday_June_06_2022_12_57_24_AM_89151835/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 663.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[`y=_G(x,y')`]
Unable to solve or complete the solution.
\[ \boxed {y^{\prime }-\frac {2 a +x^{2} \sqrt {-y^{2}+4 a x}}{y}=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x^{2} \sqrt {-y^{2}+4 a x}-y y^{\prime }+2 a =0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-\frac {-2 a -x^{2} \sqrt {-y^{2}+4 a x}}{y} \end {array} \]
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/y*(4*a*x-y^2)^(1/2)], [0, ((4*a*x-y^2)^(1/2)*x^3+12*a*x-3*y^2)/y]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 27
dsolve(diff(y(x),x) = (2*a+x^2*(-y(x)^2+4*a*x)^(1/2))/y(x),y(x), singsol=all)
\[ -\sqrt {4 a x -y \left (x \right )^{2}}-\frac {x^{3}}{3}-c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 5.05 (sec). Leaf size: 161
DSolve[y'[x] == (2*a + x^2*Sqrt[4*a*x - y[x]^2])/y[x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {\sqrt {4096 a^6 x \left (36 a-x^5\right )+128 a^3 e^{c_1} x^3-e^{2 c_1}}}{192 a^3} \\ y(x)\to \frac {\sqrt {4096 a^6 x \left (36 a-x^5\right )+128 a^3 e^{c_1} x^3-e^{2 c_1}}}{192 a^3} \\ y(x)\to -\frac {\sqrt {a^6 x \left (36 a-x^5\right )}}{3 a^3} \\ y(x)\to \frac {\sqrt {a^6 x \left (36 a-x^5\right )}}{3 a^3} \\ \end{align*}