Internal problem ID [9024]
Internal file name [OUTPUT/7959_Monday_June_06_2022_01_01_22_AM_50119571/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 690.
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.
\[ \boxed {y^{\prime }-\frac {-x^{2}+1+4 \sqrt {x^{2}+8 y-2 x +1}\, x^{3}}{4 \left (1+x \right )}=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & 4 \sqrt {x^{2}+8 y-2 x +1}\, x^{3}-4 y^{\prime } x -x^{2}-4 y^{\prime }+1=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {x^{2}-1-4 \sqrt {x^{2}+8 y-2 x +1}\, x^{3}}{-4 x -4} \end {array} \]
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))-(1/4)*(16*x^7+x^3+x^2-3*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) = (1/4)*(16*_a^7+8*_b(_a)*_a^2+_a^3+20*_b(_a)*_a+_a^2+12*_b(_a)-3*_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: 0.14 (sec). Leaf size: 40
dsolve(diff(y(x),x) = 1/4*(-x^2+1+4*x^3*(x^2-2*x+1+8*y(x))^(1/2))/(x+1),y(x), singsol=all)
\[ c_{1} +\frac {4 x^{3}}{3}-2 x^{2}-4 \ln \left (x +1\right )+4 x -\sqrt {x^{2}-2 x +1+8 y \left (x \right )} = 0 \]
✓ Solution by Mathematica
Time used: 1.35 (sec). Leaf size: 108
DSolve[y'[x] == (1/4 - x^2/4 + x^3*Sqrt[1 - 2*x + x^2 + 8*y[x]])/(1 + x),y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {2 x^6}{9}-\frac {2 x^5}{3}+\frac {11 x^4}{6}-\frac {2}{3} (3+2 c_1) x^3+\left (\frac {15}{8}+2 c_1\right ) x^2+\left (\frac {4 x^3}{3}-2 x^2+4 x-4 c_1\right ) \log \left (\frac {1}{x+1}\right )+2 \log ^2\left (\frac {1}{x+1}\right )+\left (\frac {1}{4}-4 c_1\right ) x-\frac {1}{8}+2 c_1{}^2 \]