Internal problem ID [9230]
Internal file name [OUTPUT/8166_Monday_June_06_2022_02_00_25_AM_97667580/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 897.
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.
\[ \boxed {y^{\prime }-\frac {\left (-108 y x^{\frac {3}{2}}+18 x^{\frac {9}{2}}-108 x^{\frac {3}{2}}-216 y^{3}+108 x^{3} y^{2}-18 y x^{6}+x^{9}\right ) \sqrt {x}}{-216 y+36 x^{3}-216}=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & 18 x^{5}+x^{\frac {19}{2}}-18 y x^{\frac {13}{2}}+108 y^{2} x^{\frac {7}{2}}-108 x^{2} y-36 y^{\prime } x^{3}-216 y^{3} \sqrt {x}-108 x^{2}+216 y^{\prime } y+216 y^{\prime }=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {-x^{\frac {19}{2}}+18 y x^{\frac {13}{2}}-108 y^{2} x^{\frac {7}{2}}-18 x^{5}+216 y^{3} \sqrt {x}+108 x^{2} y+108 x^{2}}{-36 x^{3}+216 y+216} \end {array} \]
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: 87
dsolve(diff(y(x),x) = (-108*x^(3/2)*y(x)+18*x^(9/2)-108*x^(3/2)-216*y(x)^3+108*x^3*y(x)^2-18*y(x)*x^6+x^9)*x^(1/2)/(-216*y(x)+36*x^3-216),y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \frac {\sqrt {9 c_{1} -12 x^{\frac {3}{2}}}\, x^{3}-3 x^{3}+18}{6 \sqrt {9 c_{1} -12 x^{\frac {3}{2}}}-18} \\ y \left (x \right ) &= \frac {\sqrt {9 c_{1} -12 x^{\frac {3}{2}}}\, x^{3}+3 x^{3}-18}{6 \sqrt {9 c_{1} -12 x^{\frac {3}{2}}}+18} \\ \end{align*}
✓ Solution by Mathematica
Time used: 2.065 (sec). Leaf size: 76
DSolve[y'[x] == (Sqrt[x]*(-108*x^(3/2) + 18*x^(9/2) + x^9 - 108*x^(3/2)*y[x] - 18*x^6*y[x] + 108*x^3*y[x]^2 - 216*y[x]^3))/(-216 + 36*x^3 - 216*y[x]),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {x^3}{6}-\frac {216}{216+\sqrt {-62208 x^{3/2}+c_1}} \\ y(x)\to \frac {x^3}{6}+\frac {216}{-216+\sqrt {-62208 x^{3/2}+c_1}} \\ y(x)\to \frac {x^3}{6} \\ \end{align*}