Internal problem ID [8953]
Internal file name [OUTPUT/7888_Monday_June_06_2022_12_50_54_AM_62856517/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 619.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[`x=_G(y,y')`]
Unable to solve or complete the solution.
\[ \boxed {y^{\prime }-\frac {6 y}{8 y^{4}+9 y^{3}+12 y^{2}+6 y-F \left (-\frac {y^{4}}{3}-\frac {y^{3}}{2}-y^{2}-y+x \right )}=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & 8 y^{\prime } y^{4}+9 y^{\prime } y^{3}+12 y^{\prime } y^{2}+6 y^{\prime } y-y^{\prime } F \left (-\frac {y^{4}}{3}-\frac {y^{3}}{2}-y^{2}-y+x \right )-6 y=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {6 y}{8 y^{4}+9 y^{3}+12 y^{2}+6 y-F \left (-\frac {y^{4}}{3}-\frac {y^{3}}{2}-y^{2}-y+x \right )} \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying homogeneous types: differential order: 1; looking for linear symmetries trying exact Looking for potential symmetries trying an equivalence to an Abel ODE 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`[0, F(-1/3*y^4-1/2*y^3-y^2-y+x)*y/(-8*y^4-9*y^3-12*y^2+F(-1/3*y^4-1/2*y^3-y^2-y
✓ Solution by Maple
Time used: 0.297 (sec). Leaf size: 81
dsolve(diff(y(x),x) = 6*y(x)/(8*y(x)^4+9*y(x)^3+12*y(x)^2+6*y(x)-F(-1/3*y(x)^4-1/2*y(x)^3-y(x)^2-y(x)+x)),y(x), singsol=all)
\[ \int _{\textit {\_b}}^{y \left (x \right )}\frac {-8 \textit {\_a}^{4}-9 \textit {\_a}^{3}-12 \textit {\_a}^{2}+F \left (-\frac {1}{3} \textit {\_a}^{4}-\frac {1}{2} \textit {\_a}^{3}-\textit {\_a}^{2}-\textit {\_a} +x \right )-6 \textit {\_a}}{F \left (-\frac {1}{3} \textit {\_a}^{4}-\frac {1}{2} \textit {\_a}^{3}-\textit {\_a}^{2}-\textit {\_a} +x \right ) \textit {\_a}}d \textit {\_a} -c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 0.583 (sec). Leaf size: 330
DSolve[y'[x] == (6*y[x])/(-F[x - y[x] - y[x]^2 - y[x]^3/2 - y[x]^4/3] + 6*y[x] + 12*y[x]^2 + 9*y[x]^3 + 8*y[x]^4),y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\int _1^{y(x)}\left (-\frac {8 K[2]^3}{F\left (-\frac {1}{3} K[2]^4-\frac {K[2]^3}{2}-K[2]^2-K[2]+x\right )}-\frac {9 K[2]^2}{F\left (-\frac {1}{3} K[2]^4-\frac {K[2]^3}{2}-K[2]^2-K[2]+x\right )}-\frac {12 K[2]}{F\left (-\frac {1}{3} K[2]^4-\frac {K[2]^3}{2}-K[2]^2-K[2]+x\right )}-\frac {F\left (-\frac {1}{3} K[2]^4-\frac {K[2]^3}{2}-K[2]^2-K[2]+x\right ) \int _1^x-\frac {6 \left (-\frac {4}{3} K[2]^3-\frac {3 K[2]^2}{2}-2 K[2]-1\right ) F'\left (-\frac {1}{3} K[2]^4-\frac {K[2]^3}{2}-K[2]^2-K[2]+K[1]\right )}{F\left (-\frac {1}{3} K[2]^4-\frac {K[2]^3}{2}-K[2]^2-K[2]+K[1]\right )^2}dK[1]+6}{F\left (-\frac {1}{3} K[2]^4-\frac {K[2]^3}{2}-K[2]^2-K[2]+x\right )}+\frac {1}{K[2]}\right )dK[2]+\int _1^x\frac {6}{F\left (-\frac {1}{3} y(x)^4-\frac {y(x)^3}{2}-y(x)^2-y(x)+K[1]\right )}dK[1]=c_1,y(x)\right ] \]