Internal problem ID [9228]
Internal file name [OUTPUT/8164_Monday_June_06_2022_01_59_59_AM_53979809/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 895.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[_rational, [_1st_order, `_with_symmetry_[F(x),G(x)]`], [_Abel, `2nd type`, `class C`]]
Unable to solve or complete the solution.
\[ \boxed {y^{\prime }-\frac {\left (-256 y a \,x^{2}-32 a^{2} x^{6}-256 a \,x^{2}+512 y^{3}+192 y^{2} a \,x^{4}+24 y a^{2} x^{8}+a^{3} x^{12}\right ) x}{512 y+64 a \,x^{4}+512}=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & -a^{3} x^{13}-24 y a^{2} x^{9}+32 a^{2} x^{7}-192 y^{2} a \,x^{5}+64 y^{\prime } a \,x^{4}+256 y a \,x^{3}-512 x y^{3}+256 x^{3} a +512 y y^{\prime }+512 y^{\prime }=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {a^{3} x^{13}+24 y a^{2} x^{9}-32 a^{2} x^{7}+192 y^{2} a \,x^{5}-256 y a \,x^{3}+512 x y^{3}-256 x^{3} a}{512 y+64 a \,x^{4}+512} \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear trying Bernoulli trying separable trying inverse linear trying homogeneous types: trying Chini differential order: 1; looking for linear symmetries trying exact trying Abel <- Abel successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 79
dsolve(diff(y(x),x) = (-256*a*x^2*y(x)-32*a^2*x^6-256*a*x^2+512*y(x)^3+192*x^4*a*y(x)^2+24*y(x)*a^2*x^8+a^3*x^12)*x/(512*y(x)+64*a*x^4+512),y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \frac {8+\left (-\sqrt {-x^{2}+c_{1}}+1\right ) a \,x^{4}}{-8+8 \sqrt {-x^{2}+c_{1}}} \\ y \left (x \right ) &= \frac {-8+\left (-\sqrt {-x^{2}+c_{1}}-1\right ) a \,x^{4}}{8+8 \sqrt {-x^{2}+c_{1}}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.474 (sec). Leaf size: 75
DSolve[y'[x] == (x*(-256*a*x^2 - 32*a^2*x^6 + a^3*x^12 - 256*a*x^2*y[x] + 24*a^2*x^8*y[x] + 192*a*x^4*y[x]^2 + 512*y[x]^3))/(512 + 64*a*x^4 + 512*y[x]),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {a x^4}{8}+\frac {512}{-512+\sqrt {-262144 x^2+c_1}} \\ y(x)\to -\frac {a x^4}{8}-\frac {512}{512+\sqrt {-262144 x^2+c_1}} \\ y(x)\to -\frac {a x^4}{8} \\ \end{align*}