Internal problem ID [9066]
Internal file name [OUTPUT/8001_Monday_June_06_2022_01_10_17_AM_64385535/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 732.
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.
\[ \boxed {y^{\prime }-\frac {-x^{2}-x -a x -a +2 \sqrt {x^{2}+2 a x +a^{2}+4 y}\, x^{3}}{2 \left (x +1\right )}=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & 2 \sqrt {x^{2}+2 a x +a^{2}+4 y}\, x^{3}-2 y^{\prime } x -a x -x^{2}-2 y^{\prime }-a -x =0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {x^{2}+x +a x +a -2 \sqrt {x^{2}+2 a x +a^{2}+4 y}\, x^{3}}{-2 x -2} \end {array} \]
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))-(1/2)*(4*x^7+2*a*x^2+x^3+5*a*x+3*x^2+3*a+ 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) = (1/2)*(4*_a^7+4*_b(_a)*_a^2+_a^3+2*_a^2*a+10*_b(_a)*_a+3*_a^2+5*_a*a+6*_b(_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: 0.156 (sec). Leaf size: 43
dsolve(diff(y(x),x) = 1/2*(-x^2-x-a*x-a+2*x^3*(x^2+2*a*x+a^2+4*y(x))^(1/2))/(x+1),y(x), singsol=all)
\[ c_{1} +\frac {2 x^{3}}{3}-x^{2}-2 \ln \left (x +1\right )+2 x -\sqrt {x^{2}+2 a x +a^{2}+4 y \left (x \right )} = 0 \]
✓ Solution by Mathematica
Time used: 1.568 (sec). Leaf size: 56
DSolve[y'[x] == (-1/2*a - x/2 - (a*x)/2 - x^2/2 + x^3*Sqrt[a^2 + 2*a*x + x^2 + 4*y[x]])/(1 + x),y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {1}{4} \left (-a^2-2 a x-x^2+\frac {1}{9} \left (-2 x^3+3 x^2-6 x+6 \log (-x-1)+6 c_1\right ){}^2\right ) \]