Internal problem ID [9046]
Internal file name [OUTPUT/7981_Monday_June_06_2022_01_07_21_AM_84206656/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 712.
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.
\[ \boxed {y^{\prime }-\frac {x^{2}+2 x +1+2 \sqrt {x^{2}+2 x +1-4 y}\, x^{3}}{2 \left (x +1\right )}=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & -2 \sqrt {x^{2}+2 x +1-4 y}\, x^{3}+2 y^{\prime } x -x^{2}+2 y^{\prime }-2 x -1=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {x^{2}+2 x +1+2 \sqrt {x^{2}+2 x +1-4 y}\, x^{3}}{2 x +2} \end {array} \]
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+x^3+5*x^2+7*x+3)/(x*(x+1)^2) 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+10*_b(_a)*_a-5*_a^2+6*_b(_a)-7*_a-3)/(_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: 38
dsolve(diff(y(x),x) = 1/2*(x^2+2*x+1+2*x^3*(x^2+2*x+1-4*y(x))^(1/2))/(x+1),y(x), singsol=all)
\[ c_{1} -\frac {2 x^{3}}{3}+x^{2}-2 x +2 \ln \left (x +1\right )-\sqrt {x^{2}+2 x +1-4 y \left (x \right )} = 0 \]
✓ Solution by Mathematica
Time used: 1.212 (sec). Leaf size: 49
DSolve[y'[x] == (1/2 + x + x^2/2 + x^3*Sqrt[1 + 2*x + x^2 - 4*y[x]])/(1 + x),y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {1}{4} \left (x^2-\frac {1}{9} \left (2 x^3-3 x^2+6 x+6 \log \left (\frac {1}{x+1}\right )-6 c_1\right ){}^2+2 x+1\right ) \]