Internal problem ID [9003]
Internal file name [OUTPUT/7938_Monday_June_06_2022_12_58_19_AM_69484223/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 669.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[[_1st_order, `_with_symmetry_[F(x),G(y)]`]]
Unable to solve or complete the solution.
\[ \boxed {y^{\prime }-\frac {\left (-2 y^{\frac {3}{2}}+3 \,{\mathrm e}^{x}\right )^{2} {\mathrm e}^{x}}{4 \sqrt {y}}=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & 4 y^{3} {\mathrm e}^{x}-12 y^{\frac {3}{2}} \left ({\mathrm e}^{x}\right )^{2}+9 \left ({\mathrm e}^{x}\right )^{3}-4 y^{\prime } \sqrt {y}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-\frac {-4 y^{3} {\mathrm e}^{x}+12 y^{\frac {3}{2}} \left ({\mathrm e}^{x}\right )^{2}-9 \left ({\mathrm e}^{x}\right )^{3}}{4 \sqrt {y}} \end {array} \]
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) = -(81/16)*y(x)*exp(4*x)-((9/2)*exp(2*x)-1)*(diff(y(x), x)), y(x)` *** Su Methods for second order ODEs: --- Trying classification methods --- trying a symmetry of the form [xi=0, eta=F(x)] checking if the LODE is missing y -> Heun: Equivalence to the GHE or one of its 4 confluent cases under a power @ Moebius -> trying a solution of the form r0(x) * Y + r1(x) * Y where Y = exp(int(r(x), dx)) * 2F1([a1, a2], [b1], f) -> Trying changes of variables to rationalize or make the ODE simpler 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 Reducible group (found an exponential solution) Reducible group (found another exponential solution) <- Kovacics algorithm successful Change of variables used: [x = ln(t)] Linear ODE actually solved: 81*t^2*u(t)+72*t*diff(u(t),t)+16*diff(diff(u(t),t),t) = 0 <- change of variables successful <- differential order: 1; linearization to 2nd order successful`
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 116
dsolve(diff(y(x),x) = 1/4*(-2*y(x)^(3/2)+3*exp(x))^2*exp(x)/y(x)^(1/2),y(x), singsol=all)
\[ \frac {\left (3 \,{\mathrm e}^{2 x -\frac {3 \,{\mathrm e}^{x}}{2}-\frac {9 \,{\mathrm e}^{2 x}}{8}}+3 c_{1} {\mathrm e}^{2 x +\frac {3 \,{\mathrm e}^{x}}{2}-\frac {9 \,{\mathrm e}^{2 x}}{8}}+2 \left (1-y \left (x \right )^{\frac {3}{2}}\right ) {\mathrm e}^{x -\frac {3 \,{\mathrm e}^{x}}{2}-\frac {9 \,{\mathrm e}^{2 x}}{8}}+2 c_{1} \left (-1-y \left (x \right )^{\frac {3}{2}}\right ) {\mathrm e}^{x +\frac {3 \,{\mathrm e}^{x}}{2}-\frac {9 \,{\mathrm e}^{2 x}}{8}}\right ) {\mathrm e}^{-x -\frac {3 \,{\mathrm e}^{x}}{2}+\frac {9 \,{\mathrm e}^{2 x}}{8}}}{-2 y \left (x \right )^{\frac {3}{2}}+3 \,{\mathrm e}^{x}-2} = 0 \]
✓ Solution by Mathematica
Time used: 60.755 (sec). Leaf size: 222
DSolve[y'[x] == (E^x*(3*E^x - 2*y[x]^(3/2))^2)/(4*Sqrt[y[x]]),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\left (-e^{3 e^x}+\frac {3}{2} e^{x+3 e^x}+\frac {3}{2} e^{x+3 c_1}+e^{3 c_1}\right ){}^{2/3}}{\sqrt [3]{\left (e^{3 e^x}+e^{3 c_1}\right ){}^2}} \\ y(x)\to -\frac {\sqrt [3]{-1} \left (-e^{3 e^x}+\frac {3}{2} e^{x+3 e^x}+\frac {3}{2} e^{x+3 c_1}+e^{3 c_1}\right ){}^{2/3}}{\sqrt [3]{\left (e^{3 e^x}+e^{3 c_1}\right ){}^2}} \\ y(x)\to \frac {\left (-\frac {1}{2}\right )^{2/3} \left (-2 e^{3 e^x}+3 e^{x+3 e^x}+3 e^{x+3 c_1}+2 e^{3 c_1}\right ){}^{2/3}}{\sqrt [3]{\left (e^{3 e^x}+e^{3 c_1}\right ){}^2}} \\ \end{align*}