Internal problem ID [9299]
Internal file name [OUTPUT/8235_Monday_June_06_2022_02_25_53_AM_54099634/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 966.
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.
\[ \boxed {y^{\prime }+\frac {1296 y}{216-1296 y-432 x y+1080 x y^{3}-2376 y^{2}-648 x^{2} y-882 y^{6}-324 y^{3} x^{2}-1944 y^{4}-1728 y^{3}+216 x^{2}+216 x^{3}+216 y^{2} x -648 y^{2} x^{2}-216 y^{4} x^{2}+216 y^{7} x -846 y^{7}+1080 y^{5} x -612 y^{5}-36 y^{11}-315 y^{9}-570 y^{8}-126 y^{10}-8 y^{12}+1152 y^{4} x +594 y^{6} x +72 x y^{8}}=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & -1296 y-216 y^{\prime }-216 x^{2} y^{\prime }+324 y^{\prime } y^{3} x^{2}+648 y^{\prime } y^{2} x^{2}-1080 y^{\prime } y^{3} x +432 y^{\prime } y x +1296 y^{\prime } y+2376 y^{\prime } y^{2}-216 y^{\prime } y^{2} x -216 y^{\prime } x^{3}+216 y^{\prime } y^{4} x^{2}+1728 y^{\prime } y^{3}+648 y^{\prime } y x^{2}-1152 y^{\prime } y^{4} x +1944 y^{\prime } y^{4}+882 y^{\prime } y^{6}+612 y^{\prime } y^{5}-1080 y^{\prime } y^{5} x +570 y^{8} y^{\prime }+846 y^{7} y^{\prime }+8 y^{12} y^{\prime }+36 y^{11} y^{\prime }+126 y^{10} y^{\prime }+315 y^{9} y^{\prime }-216 y^{7} y^{\prime } x -72 y^{8} y^{\prime } x -594 y^{6} y^{\prime } x =0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {1296 y}{-216+1296 y+432 x y-1080 x y^{3}+2376 y^{2}+648 x^{2} y+882 y^{6}+324 y^{3} x^{2}+1944 y^{4}+1728 y^{3}-216 x^{2}-216 x^{3}-216 y^{2} x +648 y^{2} x^{2}+216 y^{4} x^{2}-216 y^{7} x +846 y^{7}-1080 y^{5} x +612 y^{5}+36 y^{11}+315 y^{9}+570 y^{8}+126 y^{10}+8 y^{12}-1152 y^{4} x -594 y^{6} x -72 x y^{8}} \end {array} \]
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 = 2 `, `-> Computing symmetries using: way = 3 `, `-> Computing symmetries using: way = 4 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)] -> trying a symmetry pattern of the form [F(x),G(x)] -> trying a symmetry pattern of the form [F(y),G(y)] <- symmetry pattern of the form [F(y),G(y)] successful 1st order, trying the canonical coordinates of the invariance group -> Calling odsolve with the ODE`, diff(y(x), x) = ((4/3)*x^4+(3/2)*x^3+2*x^2+x)/x, y(x)` *** Sublevel 2 *** Methods for first order ODEs: --- Trying classification methods --- trying a quadrature <- quadrature successful <- 1st order, canonical coordinates successful`
✓ Solution by Maple
Time used: 0.11 (sec). Leaf size: 50
dsolve(diff(y(x),x) = -1296*y(x)/(216+72*y(x)^8*x+216*y(x)^7*x+1080*y(x)^5*x-882*y(x)^6-216*x^2*y(x)^4+594*x*y(x)^6+1080*x*y(x)^3-432*x*y(x)-324*x^2*y(x)^3-648*x^2*y(x)^2+1152*x*y(x)^4-648*x^2*y(x)+216*x*y(x)^2-612*y(x)^5-1944*y(x)^4-1296*y(x)-1728*y(x)^3+216*x^3-2376*y(x)^2+216*x^2-126*y(x)^10-315*y(x)^9-8*y(x)^12-36*y(x)^11-846*y(x)^7-570*y(x)^8),y(x), singsol=all)
\[ y \left (x \right ) = {\mathrm e}^{\operatorname {RootOf}\left (-\textit {\_Z} -6 \left (\int _{}^{-\frac {{\mathrm e}^{4 \textit {\_Z}}}{3}-\frac {{\mathrm e}^{3 \textit {\_Z}}}{2}-{\mathrm e}^{2 \textit {\_Z}}-{\mathrm e}^{\textit {\_Z}}+x}\frac {1}{\textit {\_a}^{3}+\textit {\_a}^{2}+1}d \textit {\_a} \right )+c_{1} \right )} \]
✓ Solution by Mathematica
Time used: 0.51 (sec). Leaf size: 292
DSolve[y'[x] == (-1296*y[x])/(216 + 216*x^2 + 216*x^3 - 1296*y[x] - 432*x*y[x] - 648*x^2*y[x] - 2376*y[x]^2 + 216*x*y[x]^2 - 648*x^2*y[x]^2 - 1728*y[x]^3 + 1080*x*y[x]^3 - 324*x^2*y[x]^3 - 1944*y[x]^4 + 1152*x*y[x]^4 - 216*x^2*y[x]^4 - 612*y[x]^5 + 1080*x*y[x]^5 - 882*y[x]^6 + 594*x*y[x]^6 - 846*y[x]^7 + 216*x*y[x]^7 - 570*y[x]^8 + 72*x*y[x]^8 - 315*y[x]^9 - 126*y[x]^10 - 36*y[x]^11 - 8*y[x]^12),y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [72 \text {RootSum}\left [-216 \text {$\#$1}^3+216 \text {$\#$1}^2 y(x)^4+324 \text {$\#$1}^2 y(x)^3+648 \text {$\#$1}^2 y(x)^2+648 \text {$\#$1}^2 y(x)-216 \text {$\#$1}^2-72 \text {$\#$1} y(x)^8-216 \text {$\#$1} y(x)^7-594 \text {$\#$1} y(x)^6-1080 \text {$\#$1} y(x)^5-1152 \text {$\#$1} y(x)^4-1080 \text {$\#$1} y(x)^3-216 \text {$\#$1} y(x)^2+432 \text {$\#$1} y(x)+8 y(x)^{12}+36 y(x)^{11}+126 y(x)^{10}+315 y(x)^9+570 y(x)^8+846 y(x)^7+882 y(x)^6+612 y(x)^5+216 y(x)^4-216 y(x)^3-216 y(x)^2-216\&,\frac {\log (x-\text {$\#$1})}{36 \text {$\#$1}^2-24 \text {$\#$1} y(x)^4-36 \text {$\#$1} y(x)^3-72 \text {$\#$1} y(x)^2-72 \text {$\#$1} y(x)+24 \text {$\#$1}+4 y(x)^8+12 y(x)^7+33 y(x)^6+60 y(x)^5+64 y(x)^4+60 y(x)^3+12 y(x)^2-24 y(x)}\&\right ]+\log (y(x))=c_1,y(x)\right ] \]