Internal problem ID [8663]
Internal file name [OUTPUT/7596_Sunday_June_05_2022_11_09_14_PM_95271948/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 327.
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.
\[ \boxed {\left (x y^{4}+2 x^{2} y^{3}+2 y+x \right ) y^{\prime }+y^{5}+y=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left (x y^{4}+2 x^{2} y^{3}+2 y+x \right ) y^{\prime }+y^{5}+y=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-\frac {y^{5}+y}{x y^{4}+2 x^{2} y^{3}+2 y+x} \end {array} \]
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 -> Calling odsolve with the ODE`, diff(diff(y(x), x), x) = -4*x^2*y(x)/(x^4+1)^2-(3*x^4-1)*(diff(y(x), x))/(x*(x^4+1)), y(x)` *** Methods for second order ODEs: --- Trying classification methods --- trying a quadrature checking if the LODE has constant coefficients checking if the LODE is of Euler type trying a symmetry of the form [xi=0, eta=F(x)] <- linear_1 successful <- differential order: 1; linearization to 2nd order successful <- change of variables {x -> y(x), y(x) -> x} succesful`
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 633
dsolve((x*y(x)^4+2*x^2*y(x)^3+2*y(x)+x)*diff(y(x),x)+y(x)^5+y(x) = 0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \frac {-1+\frac {\left (108 c_{1}^{3} x^{2}+12 \sqrt {3}\, \sqrt {27 c_{1}^{4} x^{2}+4 c_{1} x^{4}+18 c_{1}^{2} x^{2}-x^{2}-4 c_{1}}\, c_{1} x +36 c_{1} x^{2}-8\right )^{\frac {1}{3}}}{2}-\frac {2 \left (3 c_{1} x^{2}-1\right )}{\left (108 c_{1}^{3} x^{2}+12 \sqrt {3}\, \sqrt {27 c_{1}^{4} x^{2}+4 c_{1} x^{4}+18 c_{1}^{2} x^{2}-x^{2}-4 c_{1}}\, c_{1} x +36 c_{1} x^{2}-8\right )^{\frac {1}{3}}}}{3 c_{1} x} \\ y \left (x \right ) &= \frac {i \left (4-12 c_{1} x^{2}-\left (108 c_{1}^{3} x^{2}+12 \sqrt {3}\, \sqrt {27 c_{1}^{4} x^{2}+18 c_{1}^{2} x^{2}+\left (4 x^{4}-4\right ) c_{1} -x^{2}}\, c_{1} x +36 c_{1} x^{2}-8\right )^{\frac {2}{3}}\right ) \sqrt {3}+12 c_{1} x^{2}-{\left (\left (108 c_{1}^{3} x^{2}+12 \sqrt {3}\, \sqrt {27 c_{1}^{4} x^{2}+18 c_{1}^{2} x^{2}+\left (4 x^{4}-4\right ) c_{1} -x^{2}}\, c_{1} x +36 c_{1} x^{2}-8\right )^{\frac {1}{3}}+2\right )}^{2}}{12 \left (108 c_{1}^{3} x^{2}+12 \sqrt {3}\, \sqrt {27 c_{1}^{4} x^{2}+18 c_{1}^{2} x^{2}+\left (4 x^{4}-4\right ) c_{1} -x^{2}}\, c_{1} x +36 c_{1} x^{2}-8\right )^{\frac {1}{3}} x c_{1}} \\ y \left (x \right ) &= \frac {12 i \sqrt {3}\, c_{1} x^{2}+i \sqrt {3}\, \left (108 c_{1}^{3} x^{2}+12 \sqrt {3}\, \sqrt {27 c_{1}^{4} x^{2}+18 c_{1}^{2} x^{2}+\left (4 x^{4}-4\right ) c_{1} -x^{2}}\, c_{1} x +36 c_{1} x^{2}-8\right )^{\frac {2}{3}}+12 c_{1} x^{2}-4 i \sqrt {3}-\left (108 c_{1}^{3} x^{2}+12 \sqrt {3}\, \sqrt {27 c_{1}^{4} x^{2}+18 c_{1}^{2} x^{2}+\left (4 x^{4}-4\right ) c_{1} -x^{2}}\, c_{1} x +36 c_{1} x^{2}-8\right )^{\frac {2}{3}}-4 \left (108 c_{1}^{3} x^{2}+12 \sqrt {3}\, \sqrt {27 c_{1}^{4} x^{2}+18 c_{1}^{2} x^{2}+\left (4 x^{4}-4\right ) c_{1} -x^{2}}\, c_{1} x +36 c_{1} x^{2}-8\right )^{\frac {1}{3}}-4}{12 x c_{1} \left (108 c_{1}^{3} x^{2}+12 \sqrt {3}\, \sqrt {27 c_{1}^{4} x^{2}+18 c_{1}^{2} x^{2}+\left (4 x^{4}-4\right ) c_{1} -x^{2}}\, c_{1} x +36 c_{1} x^{2}-8\right )^{\frac {1}{3}}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 10.131 (sec). Leaf size: 675
DSolve[y[x] + y[x]^5 + (x + 2*y[x] + 2*x^2*y[x]^3 + x*y[x]^4)*y'[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\frac {2 c_1 \left (3 x^2+c_1\right )}{\sqrt [3]{\frac {9}{2} \left (3+c_1{}^2\right ) x^2+\frac {3}{2} \sqrt {3} \sqrt {-4 c_1{}^3 x^6+\left (27-c_1{}^4+18 c_1{}^2\right ) x^4+4 c_1{}^3 x^2}+c_1{}^3}}+2^{2/3} \sqrt [3]{9 \left (3+c_1{}^2\right ) x^2+3 \sqrt {3} \sqrt {-4 c_1{}^3 x^6+\left (27-c_1{}^4+18 c_1{}^2\right ) x^4+4 c_1{}^3 x^2}+2 c_1{}^3}+2 c_1}{6 x} \\ y(x)\to \frac {-\frac {2 i \left (\sqrt {3}-i\right ) c_1 \left (3 x^2+c_1\right )}{\sqrt [3]{\frac {9}{2} \left (3+c_1{}^2\right ) x^2+\frac {3}{2} \sqrt {3} \sqrt {-4 c_1{}^3 x^6+\left (27-c_1{}^4+18 c_1{}^2\right ) x^4+4 c_1{}^3 x^2}+c_1{}^3}}+i 2^{2/3} \left (\sqrt {3}+i\right ) \sqrt [3]{9 \left (3+c_1{}^2\right ) x^2+3 \sqrt {3} \sqrt {-4 c_1{}^3 x^6+\left (27-c_1{}^4+18 c_1{}^2\right ) x^4+4 c_1{}^3 x^2}+2 c_1{}^3}+4 c_1}{12 x} \\ y(x)\to \frac {\frac {2 i \left (\sqrt {3}+i\right ) c_1 \left (3 x^2+c_1\right )}{\sqrt [3]{\frac {9}{2} \left (3+c_1{}^2\right ) x^2+\frac {3}{2} \sqrt {3} \sqrt {-4 c_1{}^3 x^6+\left (27-c_1{}^4+18 c_1{}^2\right ) x^4+4 c_1{}^3 x^2}+c_1{}^3}}-2^{2/3} \left (1+i \sqrt {3}\right ) \sqrt [3]{9 \left (3+c_1{}^2\right ) x^2+3 \sqrt {3} \sqrt {-4 c_1{}^3 x^6+\left (27-c_1{}^4+18 c_1{}^2\right ) x^4+4 c_1{}^3 x^2}+2 c_1{}^3}+4 c_1}{12 x} \\ y(x)\to 0 \\ y(x)\to -\sqrt [4]{-1} \\ y(x)\to \sqrt [4]{-1} \\ y(x)\to -(-1)^{3/4} \\ y(x)\to (-1)^{3/4} \\ y(x)\to \frac {1}{2} x \left (-1+\frac {i x^2}{\sqrt {-x^4}}\right ) \\ y(x)\to -\frac {x}{2}+\frac {i \sqrt {-x^4}}{2 x} \\ \end{align*}