Internal problem ID [9043]
Internal file name [OUTPUT/7978_Monday_June_06_2022_01_06_48_AM_49332745/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 709.
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.
\[ \boxed {y^{\prime }-\frac {2 a x +2 a +x^{3} \sqrt {-y^{2}+4 a x}}{\left (x +1\right ) y}=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x^{3} \sqrt {-y^{2}+4 a x}-x y y^{\prime }-y^{\prime } y+2 a x +2 a =0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {-2 a x -2 a -x^{3} \sqrt {-y^{2}+4 a x}}{-x y-y} \end {array} \]
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, (4*a*x-y^2)^(1/2)/y]
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 39
dsolve(diff(y(x),x) = (2*a*x+2*a+x^3*(-y(x)^2+4*a*x)^(1/2))/(x+1)/y(x),y(x), singsol=all)
\[ -\sqrt {4 a x -y \left (x \right )^{2}}-\frac {x^{3}}{3}+\frac {x^{2}}{2}-x +\ln \left (x +1\right )-c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 3.881 (sec). Leaf size: 143
DSolve[y'[x] == (2*a + 2*a*x + x^3*Sqrt[4*a*x - y[x]^2])/((1 + x)*y[x]),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {1}{6} \sqrt {144 a x-\left (2 x^3-3 x^2+6 x+6 c_1\right ){}^2+12 \left (2 x^3-3 x^2+6 x+6 c_1\right ) \log (x+1)-36 \log ^2(x+1)} \\ y(x)\to \frac {1}{6} \sqrt {144 a x-\left (2 x^3-3 x^2+6 x+6 c_1\right ){}^2+12 \left (2 x^3-3 x^2+6 x+6 c_1\right ) \log (x+1)-36 \log ^2(x+1)} \\ \end{align*}