Internal problem ID [9303]
Internal file name [OUTPUT/8239_Monday_June_06_2022_02_27_33_AM_82601737/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 970.
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.
\[ \boxed {y^{\prime }+\frac {216 y \left (-2 y^{4}-3 y^{3}-6 y^{2}-6 y+6 x +6\right )}{1080 y^{5} x +4428 y^{5}-648 x y^{3}-432 y^{4} x -648 y^{2} x^{2}-1296 y+2484 y^{6}-1296 y^{2}+1728 y^{3}+2808 y^{4}+216 x^{3}-216 y^{4} x^{2}-1944 y^{2} x +216 y^{7} x -8 y^{12}-648 y x^{2}-1296 y x -324 y^{3} x^{2}+72 y^{8} x +594 y^{7}+594 y^{6} x -18 y^{8}-315 y^{9}-126 y^{10}-36 y^{11}}=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & 1296 y y^{\prime }+432 y^{5}-1296 y+1296 y^{2}+1296 y^{3}+648 y^{4}+1296 y^{2} y^{\prime }+1296 y^{\prime } x y-2808 y^{\prime } y^{4}+648 x^{2} y^{\prime } y-1296 y x -216 y^{\prime } x^{3}+648 y^{\prime } y^{3} x -2484 y^{\prime } y^{6}-1728 y^{\prime } y^{3}+8 y^{\prime } y^{12}+36 y^{\prime } y^{11}+126 y^{\prime } y^{10}+315 y^{\prime } y^{9}+18 y^{\prime } y^{8}-594 y^{\prime } y^{7}-4428 y^{\prime } y^{5}+324 y^{\prime } y^{3} x^{2}+648 y^{\prime } y^{2} x^{2}+1944 y^{\prime } y^{2} x +432 y^{\prime } y^{4} x -72 y^{\prime } y^{8} x +216 y^{\prime } y^{4} x^{2}-216 y^{\prime } y^{7} x -594 y^{\prime } y^{6} x -1080 y^{\prime } y^{5} x =0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {-432 y^{5}+1296 y-1296 y^{2}-1296 y^{3}-648 y^{4}+1296 y x}{-1080 y^{5} x -4428 y^{5}+648 x y^{3}+432 y^{4} x +648 y^{2} x^{2}+1296 y-2484 y^{6}+1296 y^{2}-1728 y^{3}-2808 y^{4}-216 x^{3}+216 y^{4} x^{2}+1944 y^{2} x -216 y^{7} x +8 y^{12}+648 y x^{2}+1296 y x +324 y^{3} x^{2}-72 y^{8} x -594 y^{7}-594 y^{6} x +18 y^{8}+315 y^{9}+126 y^{10}+36 y^{11}} \end {array} \]
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 <- 1st order, canonical coordinates successful`
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 181
dsolve(diff(y(x),x) = -216*y(x)*(-2*y(x)^4-3*y(x)^3-6*y(x)^2-6*y(x)+6*x+6)/(72*y(x)^8*x+216*y(x)^7*x+1080*y(x)^5*x+2484*y(x)^6-216*x^2*y(x)^4+594*x*y(x)^6-648*x*y(x)^3-1296*x*y(x)-324*x^2*y(x)^3-648*x^2*y(x)^2-432*x*y(x)^4-648*x^2*y(x)-1944*x*y(x)^2+4428*y(x)^5+2808*y(x)^4-1296*y(x)+1728*y(x)^3+216*x^3-1296*y(x)^2-126*y(x)^10-315*y(x)^9-8*y(x)^12-36*y(x)^11+594*y(x)^7-18*y(x)^8),y(x), singsol=all)
\begin{align*} \frac {-6 \sqrt {3 \ln \left (y \left (x \right )\right )-108 c_{1} +9}+\left (2 y \left (x \right )^{4}+3 y \left (x \right )^{3}+6 y \left (x \right )^{2}-6 x +6 y \left (x \right )\right ) \ln \left (y \left (x \right )\right )-72 y \left (x \right )^{4} c_{1} -108 c_{1} y \left (x \right )^{3}-216 c_{1} y \left (x \right )^{2}+216 c_{1} x -216 c_{1} y \left (x \right )+18}{216 c_{1} -6 \ln \left (y \left (x \right )\right )} &= 0 \\ \frac {6 \sqrt {3 \ln \left (y \left (x \right )\right )-108 c_{1} +9}+\left (2 y \left (x \right )^{4}+3 y \left (x \right )^{3}+6 y \left (x \right )^{2}-6 x +6 y \left (x \right )\right ) \ln \left (y \left (x \right )\right )-72 y \left (x \right )^{4} c_{1} -108 c_{1} y \left (x \right )^{3}-216 c_{1} y \left (x \right )^{2}+216 c_{1} x -216 c_{1} y \left (x \right )+18}{216 c_{1} -6 \ln \left (y \left (x \right )\right )} &= 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.451 (sec). Leaf size: 66
DSolve[y'[x] == (-216*y[x]*(6 + 6*x - 6*y[x] - 6*y[x]^2 - 3*y[x]^3 - 2*y[x]^4))/(216*x^3 - 1296*y[x] - 1296*x*y[x] - 648*x^2*y[x] - 1296*y[x]^2 - 1944*x*y[x]^2 - 648*x^2*y[x]^2 + 1728*y[x]^3 - 648*x*y[x]^3 - 324*x^2*y[x]^3 + 2808*y[x]^4 - 432*x*y[x]^4 - 216*x^2*y[x]^4 + 4428*y[x]^5 + 1080*x*y[x]^5 + 2484*y[x]^6 + 594*x*y[x]^6 + 594*y[x]^7 + 216*x*y[x]^7 - 18*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]
\[ \text {Solve}\left [\frac {36 \left (2 y(x)^4+3 y(x)^3+6 y(x)^2+6 y(x)-6 x-3\right )}{\left (y(x) \left (2 y(x)^3+3 y(x)^2+6 y(x)+6\right )-6 x\right )^2}+\log (y(x))=c_1,y(x)\right ] \]