Internal problem ID [9033]
Internal file name [OUTPUT/7968_Monday_June_06_2022_01_02_48_AM_56149969/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 699.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[[_1st_order, `_with_symmetry_[F(x),G(x)]`]]
Unable to solve or complete the solution.
\[ \boxed {y^{\prime }-\frac {x \left (-2 x -2+3 x^{2} \sqrt {x^{2}+3 y}\right )}{3 \left (x +1\right )}=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & 3 \sqrt {x^{2}+3 y}\, x^{3}-3 y^{\prime } x -2 x^{2}-3 y^{\prime }-2 x =0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {-3 \sqrt {x^{2}+3 y}\, x^{3}+2 x^{2}+2 x}{-3 x -3} \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying homogeneous types: differential order: 1; looking for linear symmetries trying exact Looking for potential symmetries trying an equivalence to an Abel ODE trying 1st order ODE linearizable_by_differentiation -> Calling odsolve with the ODE`, diff(diff(y(x), x), x)-(2*x+3)*(diff(y(x), x))/(x*(x+1))-(1/6)*(9*x^6+4*x^2+12*x+8)/(x+1)^2, y(x)` Methods for second order ODEs: --- Trying classification methods --- trying a quadrature trying high order exact linear fully integrable -> Calling odsolve with the ODE`, diff(_b(_a), _a) = (1/6)*(9*_a^7+12*_b(_a)*_a^2+4*_a^3+30*_b(_a)*_a+12*_a^2+18*_b(_a)+8*_a)/(_a Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear <- 1st order linear successful <- high order exact linear fully integrable successful <- 1st order ODE linearizable_by_differentiation successful`
✓ Solution by Maple
Time used: 0.14 (sec). Leaf size: 36
dsolve(diff(y(x),x) = 1/3*x*(-2*x-2+3*x^2*(x^2+3*y(x))^(1/2))/(x+1),y(x), singsol=all)
\[ c_{1} +\frac {x^{3}}{2}-\frac {3 x^{2}}{4}-\frac {3 \ln \left (x +1\right )}{2}+\frac {3 x}{2}-\sqrt {x^{2}+3 y \left (x \right )} = 0 \]
✓ Solution by Mathematica
Time used: 0.915 (sec). Leaf size: 47
DSolve[y'[x] == (x*(-2 - 2*x + 3*x^2*Sqrt[x^2 + 3*y[x]]))/(3*(1 + x)),y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {1}{3} \left (-x^2+\frac {1}{16} \left (2 x^3-3 x^2+6 x+6 \log \left (\frac {1}{x+1}\right )-6 c_1\right ){}^2\right ) \]