Internal problem ID [8968]
Internal file name [OUTPUT/7903_Monday_June_06_2022_12_52_57_AM_11856338/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 634.
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)*y+H(x)]`]]
Unable to solve or complete the solution.
\[ \boxed {y^{\prime }-\frac {1+2 x^{5} \sqrt {1+4 x^{2} y}}{2 x^{3}}=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & 2 x^{5} \sqrt {1+4 x^{2} y}-2 y^{\prime } x^{3}+1=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-\frac {-1-2 x^{5} \sqrt {1+4 x^{2} y}}{2 x^{3}} \end {array} \]
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)-3*(diff(y(x), x))/x-(2*x^10-3)/x^4, y(x)` *** Sublevel 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) = (2*_a^10+3*_a^3*_b(_a)-3)/_a^4, _b(_a)` *** Sublevel 3 *** 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: 0.157 (sec). Leaf size: 29
dsolve(diff(y(x),x) = 1/2*(1+2*x^5*(4*x^2*y(x)+1)^(1/2))/x^3,y(x), singsol=all)
\[ \frac {x^{5}+2 c_{1} x -2 \sqrt {4 x^{2} y \left (x \right )+1}}{2 x} = 0 \]
✓ Solution by Mathematica
Time used: 0.393 (sec). Leaf size: 31
DSolve[y'[x] == (1/2 + x^5*Sqrt[1 + 4*x^2*y[x]])/x^3,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {1}{16} \left (x^8-8 c_1 x^4-\frac {4}{x^2}+16 c_1{}^2\right ) \]