Internal problem ID [9251]
Internal file name [OUTPUT/8187_Monday_June_06_2022_02_08_14_AM_97893892/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 918.
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.
\[ \boxed {y^{\prime }-\frac {2 y^{8}}{y^{5}+2 y^{6}+2 y^{2}+16 y^{4} x +32 y^{6} x^{2}+2+24 y^{2} x +96 y^{4} x^{2}+128 y^{6} x^{3}}=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & 128 y^{\prime } y^{6} x^{3}+32 y^{\prime } y^{6} x^{2}-2 y^{8}+2 y^{\prime } y^{6}+96 y^{\prime } y^{4} x^{2}+y^{\prime } y^{5}+16 y^{\prime } y^{4} x +24 y^{\prime } y^{2} x +2 y^{\prime } y^{2}+2 y^{\prime }=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {2 y^{8}}{y^{5}+2 y^{6}+2 y^{2}+16 y^{4} x +32 y^{6} x^{2}+2+24 y^{2} x +96 y^{4} x^{2}+128 y^{6} x^{3}} \end {array} \]
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: 0.046 (sec). Leaf size: 41
dsolve(diff(y(x),x) = 2*y(x)^8/(y(x)^5+2*y(x)^6+2*y(x)^2+16*x*y(x)^4+32*y(x)^6*x^2+2+24*x*y(x)^2+96*x^2*y(x)^4+128*x^3*y(x)^6),y(x), singsol=all)
\[ x -\operatorname {RootOf}\left (\left (\int _{}^{\textit {\_Z}}\frac {1}{64 \textit {\_a}^{3}+16 \textit {\_a}^{2}+1}d \textit {\_a} \right ) y \left (x \right )+c_{1} y \left (x \right )+1\right )+\frac {1}{4 y \left (x \right )^{2}} = 0 \]
✓ Solution by Mathematica
Time used: 0.351 (sec). Leaf size: 720
DSolve[y'[x] == (2*y[x]^8)/(2 + 2*y[x]^2 + 24*x*y[x]^2 + 16*x*y[x]^4 + 96*x^2*y[x]^4 + y[x]^5 + 2*y[x]^6 + 32*x^2*y[x]^6 + 128*x^3*y[x]^6),y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\int _1^{y(x)}\left (\text {RootSum}\left [64 \text {$\#$1}^3 K[1]^6+16 \text {$\#$1}^2 K[1]^6+K[1]^6+48 \text {$\#$1}^2 K[1]^4+8 \text {$\#$1} K[1]^4+12 \text {$\#$1} K[1]^2+K[1]^2+1\&,\frac {\log (x-\text {$\#$1})}{48 \text {$\#$1}^2 K[1]^4+8 \text {$\#$1} K[1]^4+24 \text {$\#$1} K[1]^2+2 K[1]^2+3}\&\right ] K[1]^3+\frac {K[1]^3}{2 \left (64 x^3 K[1]^6+16 x^2 K[1]^6+K[1]^6+48 x^2 K[1]^4+8 x K[1]^4+12 x K[1]^2+K[1]^2+1\right )}-\frac {\text {RootSum}\left [64 \text {$\#$1}^3 K[1]^6+16 \text {$\#$1}^2 K[1]^6+K[1]^6+48 \text {$\#$1}^2 K[1]^4+8 \text {$\#$1} K[1]^4+12 \text {$\#$1} K[1]^2+K[1]^2+1\&,\frac {128 x \log (x-\text {$\#$1}) \text {$\#$1}^2 K[1]^6+320 \log (x-\text {$\#$1}) \text {$\#$1}^2 K[1]^6-24 x \log (x-\text {$\#$1}) K[1]^6+2 \log (x-\text {$\#$1}) K[1]^6-288 x \log (x-\text {$\#$1}) \text {$\#$1} K[1]^6+24 \log (x-\text {$\#$1}) \text {$\#$1} K[1]^6+32 \log (x-\text {$\#$1}) \text {$\#$1}^2 K[1]^4+16 \text {$\#$1}^2 K[1]^4-72 x \log (x-\text {$\#$1}) K[1]^4+64 x \log (x-\text {$\#$1}) \text {$\#$1} K[1]^4+88 \log (x-\text {$\#$1}) \text {$\#$1} K[1]^4-36 \text {$\#$1} K[1]^4-3 K[1]^4+8 x \log (x-\text {$\#$1}) K[1]^2+2 \log (x-\text {$\#$1}) K[1]^2+16 \log (x-\text {$\#$1}) \text {$\#$1} K[1]^2+8 \text {$\#$1} K[1]^2-9 K[1]^2+2 \log (x-\text {$\#$1})+1}{64 x \text {$\#$1}^2 K[1]^6+160 \text {$\#$1}^2 K[1]^6+112 x K[1]^6-144 x \text {$\#$1} K[1]^6-112 \text {$\#$1} K[1]^6+K[1]^6+16 \text {$\#$1}^2 K[1]^4-36 x K[1]^4+32 x \text {$\#$1} K[1]^4+44 \text {$\#$1} K[1]^4+4 x K[1]^2+8 \text {$\#$1} K[1]^2+K[1]^2+1}\&\right ]}{2 K[1]}+\frac {1}{K[1]^2}\right )dK[1]-\frac {1}{4} y(x)^4 \text {RootSum}\left [64 \text {$\#$1}^3 y(x)^6+16 \text {$\#$1}^2 y(x)^6+48 \text {$\#$1}^2 y(x)^4+8 \text {$\#$1} y(x)^4+12 \text {$\#$1} y(x)^2+y(x)^6+y(x)^2+1\&,\frac {\log (x-\text {$\#$1})}{48 \text {$\#$1}^2 y(x)^4+8 \text {$\#$1} y(x)^4+24 \text {$\#$1} y(x)^2+2 y(x)^2+3}\&\right ]=c_1,y(x)\right ] \]