Internal problem ID [9253]
Internal file name [OUTPUT/8189_Monday_June_06_2022_02_08_32_AM_33551611/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 920.
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.
\[ \boxed {y^{\prime }-\frac {2 y^{6} \left (1+4 y^{2} x +y^{2}\right )}{y^{3}+4 y^{5} x +y^{5}+2+24 y^{2} x +96 y^{4} x^{2}+128 y^{6} x^{3}}=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & 128 y^{\prime } y^{6} x^{3}-8 y^{8} x -2 y^{8}+4 y^{\prime } y^{5} x +96 y^{\prime } y^{4} x^{2}+y^{\prime } y^{5}-2 y^{6}+y^{\prime } y^{3}+24 y^{\prime } y^{2} x +2 y^{\prime }=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {8 y^{8} x +2 y^{8}+2 y^{6}}{y^{3}+4 y^{5} x +y^{5}+2+24 y^{2} x +96 y^{4} x^{2}+128 y^{6} x^{3}} \end {array} \]
✗ Solution by Maple
dsolve(diff(y(x),x) = 2*y(x)^6*(1+4*x*y(x)^2+y(x)^2)/(y(x)^3+4*y(x)^5*x+y(x)^5+2+24*x*y(x)^2+96*x^2*y(x)^4+128*x^3*y(x)^6),y(x), singsol=all)
\[ \text {No solution found} \]
✓ Solution by Mathematica
Time used: 5.894 (sec). Leaf size: 301
DSolve[y'[x] == (2*y[x]^6*(1 + y[x]^2 + 4*x*y[x]^2))/(2 + 24*x*y[x]^2 + y[x]^3 + 96*x^2*y[x]^4 + y[x]^5 + 4*x*y[x]^5 + 128*x^3*y[x]^6),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \text {Root}\left [\text {$\#$1}^5 \left (128 c_1 x^2-8 x-1\right )+128 \text {$\#$1}^4 x^2+\text {$\#$1}^3 (-2+64 c_1 x)+64 \text {$\#$1}^2 x+8 \text {$\#$1} c_1+8\&,1\right ] \\ y(x)\to \text {Root}\left [\text {$\#$1}^5 \left (128 c_1 x^2-8 x-1\right )+128 \text {$\#$1}^4 x^2+\text {$\#$1}^3 (-2+64 c_1 x)+64 \text {$\#$1}^2 x+8 \text {$\#$1} c_1+8\&,2\right ] \\ y(x)\to \text {Root}\left [\text {$\#$1}^5 \left (128 c_1 x^2-8 x-1\right )+128 \text {$\#$1}^4 x^2+\text {$\#$1}^3 (-2+64 c_1 x)+64 \text {$\#$1}^2 x+8 \text {$\#$1} c_1+8\&,3\right ] \\ y(x)\to \text {Root}\left [\text {$\#$1}^5 \left (128 c_1 x^2-8 x-1\right )+128 \text {$\#$1}^4 x^2+\text {$\#$1}^3 (-2+64 c_1 x)+64 \text {$\#$1}^2 x+8 \text {$\#$1} c_1+8\&,4\right ] \\ y(x)\to \text {Root}\left [\text {$\#$1}^5 \left (128 c_1 x^2-8 x-1\right )+128 \text {$\#$1}^4 x^2+\text {$\#$1}^3 (-2+64 c_1 x)+64 \text {$\#$1}^2 x+8 \text {$\#$1} c_1+8\&,5\right ] \\ \end{align*}