Internal problem ID [8981]
Internal file name [OUTPUT/7916_Monday_June_06_2022_12_55_00_AM_29894122/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 647.
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(y)]`]]
Unable to solve or complete the solution.
\[ \boxed {y^{\prime }-\frac {\left (y^{2} a +b \,x^{2}\right )^{2} x}{a^{\frac {5}{2}} y}=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime } a^{\frac {5}{2}} y-y^{4} a^{2} x -2 y^{2} a b \,x^{3}-b^{2} x^{5}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {y^{4} a^{2} x +2 y^{2} a b \,x^{3}+b^{2} x^{5}}{a^{\frac {5}{2}} y} \end {array} \]
Maple trace Kovacic algorithm successful
`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 Looking for potential symmetries trying inverse_Riccati trying an equivalence to an Abel ODE differential order: 1; trying a linearization to 2nd order -> Calling odsolve with the ODE`, diff(diff(y(x), x), x) = -4*b^2*x^6*y(x)/a^3-(-4*b*x^4*a^(1/2)-a^2)*(diff(y(x), x))/(a^2*x), y(x)` Methods for second order ODEs: --- Trying classification methods --- trying a quadrature checking if the LODE has constant coefficients checking if the LODE is of Euler type trying a symmetry of the form [xi=0, eta=F(x)] checking if the LODE is missing y -> Trying a Liouvillian solution using Kovacics algorithm A Liouvillian solution exists Group is reducible or imprimitive <- Kovacics algorithm successful <- differential order: 1; linearization to 2nd order successful`
✓ Solution by Maple
Time used: 0.032 (sec). Leaf size: 460
dsolve(diff(y(x),x) = (a*y(x)^2+b*x^2)^2*x/a^(5/2)/y(x),y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \frac {\sqrt {-\left (\left (b \,x^{2}-\sqrt {-\frac {b}{a^{\frac {3}{2}}}}\, a^{\frac {3}{2}}\right ) {\mathrm e}^{\frac {x^{2} \left (-2 \sqrt {-\frac {b}{a^{\frac {3}{2}}}}\, a^{\frac {3}{2}}+b \,x^{2}\right )}{2 a^{\frac {3}{2}}}}+c_{1} \left (\sqrt {-\frac {b}{a^{\frac {3}{2}}}}\, a^{\frac {3}{2}}+b \,x^{2}\right ) {\mathrm e}^{\frac {x^{2} \left (2 \sqrt {-\frac {b}{a^{\frac {3}{2}}}}\, a^{\frac {3}{2}}+b \,x^{2}\right )}{2 a^{\frac {3}{2}}}}\right ) \left (c_{1} {\mathrm e}^{\frac {x^{2} \left (2 \sqrt {-\frac {b}{a^{\frac {3}{2}}}}\, a^{\frac {3}{2}}+b \,x^{2}\right )}{2 a^{\frac {3}{2}}}}+{\mathrm e}^{\frac {x^{2} \left (-2 \sqrt {-\frac {b}{a^{\frac {3}{2}}}}\, a^{\frac {3}{2}}+b \,x^{2}\right )}{2 a^{\frac {3}{2}}}}\right ) a}}{a \left (c_{1} {\mathrm e}^{\frac {x^{2} \left (2 \sqrt {-\frac {b}{a^{\frac {3}{2}}}}\, a^{\frac {3}{2}}+b \,x^{2}\right )}{2 a^{\frac {3}{2}}}}+{\mathrm e}^{\frac {x^{2} \left (-2 \sqrt {-\frac {b}{a^{\frac {3}{2}}}}\, a^{\frac {3}{2}}+b \,x^{2}\right )}{2 a^{\frac {3}{2}}}}\right )} \\ y \left (x \right ) &= -\frac {\sqrt {-\left (\left (b \,x^{2}-\sqrt {-\frac {b}{a^{\frac {3}{2}}}}\, a^{\frac {3}{2}}\right ) {\mathrm e}^{\frac {x^{2} \left (-2 \sqrt {-\frac {b}{a^{\frac {3}{2}}}}\, a^{\frac {3}{2}}+b \,x^{2}\right )}{2 a^{\frac {3}{2}}}}+c_{1} \left (\sqrt {-\frac {b}{a^{\frac {3}{2}}}}\, a^{\frac {3}{2}}+b \,x^{2}\right ) {\mathrm e}^{\frac {x^{2} \left (2 \sqrt {-\frac {b}{a^{\frac {3}{2}}}}\, a^{\frac {3}{2}}+b \,x^{2}\right )}{2 a^{\frac {3}{2}}}}\right ) \left (c_{1} {\mathrm e}^{\frac {x^{2} \left (2 \sqrt {-\frac {b}{a^{\frac {3}{2}}}}\, a^{\frac {3}{2}}+b \,x^{2}\right )}{2 a^{\frac {3}{2}}}}+{\mathrm e}^{\frac {x^{2} \left (-2 \sqrt {-\frac {b}{a^{\frac {3}{2}}}}\, a^{\frac {3}{2}}+b \,x^{2}\right )}{2 a^{\frac {3}{2}}}}\right ) a}}{a \left (c_{1} {\mathrm e}^{\frac {x^{2} \left (2 \sqrt {-\frac {b}{a^{\frac {3}{2}}}}\, a^{\frac {3}{2}}+b \,x^{2}\right )}{2 a^{\frac {3}{2}}}}+{\mathrm e}^{\frac {x^{2} \left (-2 \sqrt {-\frac {b}{a^{\frac {3}{2}}}}\, a^{\frac {3}{2}}+b \,x^{2}\right )}{2 a^{\frac {3}{2}}}}\right )} \\ \end{align*}
✓ Solution by Mathematica
Time used: 14.816 (sec). Leaf size: 117
DSolve[y'[x] == (x*(b*x^2 + a*y[x]^2)^2)/(a^(5/2)*y[x]),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\sqrt {\frac {-b x^2+a^{3/4} \sqrt {b} \tan \left (\frac {a^{3/2} b x^2+2 c_1}{a^{9/4} \sqrt {b}}\right )}{a}} \\ y(x)\to \sqrt {\frac {-b x^2+a^{3/4} \sqrt {b} \tan \left (\frac {a^{3/2} b x^2+2 c_1}{a^{9/4} \sqrt {b}}\right )}{a}} \\ \end{align*}