Internal problem ID [9222]
Internal file name [OUTPUT/8158_Monday_June_06_2022_01_58_54_AM_86627300/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 889.
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.
\[ \boxed {y^{\prime }+\frac {\left (-8-8 y^{3}+24 \,{\mathrm e}^{x} y^{{3}/{2}}-18 \,{\mathrm e}^{2 x}-8 y^{{9}/{2}}+36 y^{3} {\mathrm e}^{x}-54 y^{{3}/{2}} {\mathrm e}^{2 x}+27 \,{\mathrm e}^{3 x}\right ) {\mathrm e}^{x}}{8 \sqrt {y}}=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & -8 y^{{9}/{2}} {\mathrm e}^{x}-54 y^{{3}/{2}} \left ({\mathrm e}^{x}\right )^{3}+36 y^{3} \left ({\mathrm e}^{x}\right )^{2}+24 y^{{3}/{2}} \left ({\mathrm e}^{x}\right )^{2}-8 y^{3} {\mathrm e}^{x}+27 \left ({\mathrm e}^{x}\right )^{4}-18 \left ({\mathrm e}^{x}\right )^{3}+8 y^{\prime } \sqrt {y}-8 \,{\mathrm e}^{x}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {8 y^{{9}/{2}} {\mathrm e}^{x}+54 y^{{3}/{2}} \left ({\mathrm e}^{x}\right )^{3}-36 y^{3} \left ({\mathrm e}^{x}\right )^{2}-24 y^{{3}/{2}} \left ({\mathrm e}^{x}\right )^{2}+8 y^{3} {\mathrm e}^{x}-27 \left ({\mathrm e}^{x}\right )^{4}+18 \left ({\mathrm e}^{x}\right )^{3}+8 \,{\mathrm e}^{x}}{8 \sqrt {y}} \end {array} \]
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 Looking for potential symmetries trying inverse_Riccati trying an equivalence to an Abel ODE differential order: 1; trying a linearization to 2nd order --- trying a change of variables {x -> y(x), y(x) -> x} differential order: 1; trying a linearization to 2nd order trying 1st order ODE linearizable_by_differentiation --- Trying Lie symmetry methods, 1st order --- `, `-> Computing symmetries using: way = 3 `, `-> Computing symmetries using: way = 4 `, `-> Computing symmetries using: way = 5 trying symmetry patterns for 1st order ODEs -> trying a symmetry pattern of the form [F(x)*G(y), 0] -> trying a symmetry pattern of the form [0, F(x)*G(y)] -> trying symmetry patterns of the forms [F(x),G(y)] and [G(y),F(x)] `, `-> Computing symmetries using: way = HINT 1st order, trying the canonical coordinates of the invariance group <- 1st order, canonical coordinates successful <- symmetry patterns of the forms [F(x),G(y)] and [G(y),F(x)] successful`
✓ Solution by Maple
Time used: 0.125 (sec). Leaf size: 92
dsolve(diff(y(x),x) = -1/8*(-8-8*y(x)^3+24*y(x)^(3/2)*exp(x)-18*exp(x)^2-8*y(x)^(9/2)+36*y(x)^3*exp(x)-54*y(x)^(3/2)*exp(x)^2+27*exp(x)^3)*exp(x)/y(x)^(1/2),y(x), singsol=all)
\[ \frac {\left (-6 \,{\mathrm e}^{x}+4 y \left (x \right )^{\frac {3}{2}}\right ) \ln \left (2 y \left (x \right )^{\frac {3}{2}}-3 \,{\mathrm e}^{x}+2\right )+\left (6 \,{\mathrm e}^{x}-4 y \left (x \right )^{\frac {3}{2}}\right ) \ln \left (2 y \left (x \right )^{\frac {3}{2}}-3 \,{\mathrm e}^{x}\right )+\left (6 c_{1} -6 \,{\mathrm e}^{x}\right ) y \left (x \right )^{\frac {3}{2}}-9 c_{1} {\mathrm e}^{x}+9 \,{\mathrm e}^{2 x}-4}{-6 y \left (x \right )^{\frac {3}{2}}+9 \,{\mathrm e}^{x}} = 0 \]
✓ Solution by Mathematica
Time used: 1.133 (sec). Leaf size: 68
DSolve[y'[x] == -1/8*(E^x*(-8 - 18*E^(2*x) + 27*E^(3*x) + 24*E^x*y[x]^(3/2) - 54*E^(2*x)*y[x]^(3/2) - 8*y[x]^3 + 36*E^x*y[x]^3 - 8*y[x]^(9/2)))/Sqrt[y[x]],y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\frac {2}{3} \log \left (y(x)^{3/2}-\frac {3 e^x}{2}\right )+e^x=\frac {4}{9 e^x-6 y(x)^{3/2}}+\frac {2}{3} \log \left (y(x)^{3/2}-\frac {3 e^x}{2}+1\right )+c_1,y(x)\right ] \]