Internal problem ID [9064]
Internal file name [OUTPUT/7999_Monday_June_06_2022_01_10_03_AM_80074697/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 730.
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.
\[ \boxed {y^{\prime }-\frac {\left (2 y^{\frac {3}{2}}-3 \,{\mathrm e}^{x}\right )^{3} {\mathrm e}^{x}}{4 \left (2 y^{\frac {3}{2}}-3 \,{\mathrm e}^{x}+2\right ) \sqrt {y}}=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & -8 y^{\frac {9}{2}} {\mathrm e}^{x}+36 y^{3} \left ({\mathrm e}^{x}\right )^{2}-54 y^{\frac {3}{2}} \left ({\mathrm e}^{x}\right )^{3}+8 y^{\prime } y^{2}+27 \left ({\mathrm e}^{x}\right )^{4}-12 y^{\prime } \sqrt {y}\, {\mathrm e}^{x}+8 y^{\prime } \sqrt {y}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {8 y^{\frac {9}{2}} {\mathrm e}^{x}-36 y^{3} \left ({\mathrm e}^{x}\right )^{2}+54 y^{\frac {3}{2}} \left ({\mathrm e}^{x}\right )^{3}-27 \left ({\mathrm e}^{x}\right )^{4}}{8 y^{2}-12 \,{\mathrm e}^{x} \sqrt {y}+8 \sqrt {y}} \end {array} \]
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 -> Calling odsolve with the ODE`, y(x)+diff(y(x), x), y(x)` *** Sublevel 2 *** Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear <- 1st order linear successful 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: 0.141 (sec). Leaf size: 37
dsolve(diff(y(x),x) = 1/4*(2*y(x)^(3/2)-3*exp(x))^3*exp(x)/(2*y(x)^(3/2)-3*exp(x)+2)/y(x)^(1/2),y(x), singsol=all)
\[ {\mathrm e}^{x}-\frac {2 \left (\int _{}^{y \left (x \right )^{\frac {3}{2}}-\frac {3 \,{\mathrm e}^{x}}{2}}\frac {\textit {\_a} +1}{\textit {\_a}^{3}-\textit {\_a} -1}d \textit {\_a} \right )}{3}-c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 0.222 (sec). Leaf size: 83
DSolve[y'[x] == (E^x*(-3*E^x + 2*y[x]^(3/2))^3)/(4*Sqrt[y[x]]*(2 - 3*E^x + 2*y[x]^(3/2))),y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [-\frac {2}{3} \text {RootSum}\left [\text {$\#$1}^3-\text {$\#$1}-1\&,\frac {\text {$\#$1} \log \left (-\text {$\#$1}+y(x)^{3/2}-\frac {3 e^x}{2}\right )+\log \left (-\text {$\#$1}+y(x)^{3/2}-\frac {3 e^x}{2}\right )}{3 \text {$\#$1}^2-1}\&\right ]+e^x-c_1=0,y(x)\right ] \]