Internal problem ID [9281]
Internal file name [OUTPUT/8217_Monday_June_06_2022_02_19_46_AM_7693270/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 948.
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]
Unable to solve or complete the solution.
\[ \boxed {y^{\prime }+\frac {216 y}{-216 y^{4}-252 y^{3}-396 y^{2}-216 y+36 x^{2}-72 y x +60 y^{5}-36 x y^{3}-72 y^{2} x -24 y^{4} x +4 y^{8}+12 y^{7}+33 y^{6}}=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & 4 y^{\prime } y^{8}+12 y^{\prime } y^{7}+33 y^{\prime } y^{6}+60 y^{\prime } y^{5}-24 y^{\prime } y^{4} x -216 y^{\prime } y^{4}-36 y^{\prime } y^{3} x -252 y^{\prime } y^{3}-72 y^{\prime } y^{2} x -396 y^{2} y^{\prime }-72 y^{\prime } x y+36 x^{2} y^{\prime }-216 y y^{\prime }+216 y=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-\frac {216 y}{-216 y^{4}-252 y^{3}-396 y^{2}-216 y+36 x^{2}-72 y x +60 y^{5}-36 x y^{3}-72 y^{2} x -24 y^{4} x +4 y^{8}+12 y^{7}+33 y^{6}} \end {array} \]
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 Riccati sub-methods: <- Riccati particular case Kamke (d) successful <- inverse_Riccati successful`
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 68
dsolve(diff(y(x),x) = -216*y(x)/(-216*y(x)^4-252*y(x)^3-396*y(x)^2-216*y(x)+36*x^2-72*x*y(x)+60*y(x)^5-36*x*y(x)^3-72*x*y(x)^2-24*x*y(x)^4+4*y(x)^8+12*y(x)^7+33*y(x)^6),y(x), singsol=all)
\[ y \left (x \right ) = {\mathrm e}^{\operatorname {RootOf}\left (12 c_{1} {\mathrm e}^{4 \textit {\_Z}}+2 \,{\mathrm e}^{4 \textit {\_Z}} \textit {\_Z} +18 c_{1} {\mathrm e}^{3 \textit {\_Z}}+3 \,{\mathrm e}^{3 \textit {\_Z}} \textit {\_Z} +36 \,{\mathrm e}^{2 \textit {\_Z}} c_{1} +6 \textit {\_Z} \,{\mathrm e}^{2 \textit {\_Z}}+36 c_{1} {\mathrm e}^{\textit {\_Z}}+6 \,{\mathrm e}^{\textit {\_Z}} \textit {\_Z} -36 c_{1} x -6 x \textit {\_Z} +36\right )} \]
✓ Solution by Mathematica
Time used: 0.464 (sec). Leaf size: 39
DSolve[y'[x] == (-216*y[x])/(36*x^2 - 216*y[x] - 72*x*y[x] - 396*y[x]^2 - 72*x*y[x]^2 - 252*y[x]^3 - 36*x*y[x]^3 - 216*y[x]^4 - 24*x*y[x]^4 + 60*y[x]^5 + 33*y[x]^6 + 12*y[x]^7 + 4*y[x]^8),y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\frac {36}{y(x) \left (2 y(x)^3+3 y(x)^2+6 y(x)+6\right )-6 x}+\log (y(x))=c_1,y(x)\right ] \]