Internal problem ID [8975]
Internal file name [OUTPUT/7910_Monday_June_06_2022_12_54_08_AM_70176800/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 641.
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.
\[ \boxed {y^{\prime }-\frac {1+2 \sqrt {4 x^{2} y+1}\, x^{4}}{2 x^{3}}=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & -2 \sqrt {4 x^{2} y+1}\, x^{4}+2 y^{\prime } x^{3}-1=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {1+2 \sqrt {4 x^{2} y+1}\, x^{4}}{2 x^{3}} \end {array} \]
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)-2*(diff(y(x), x))/x-(1/2)*(4*x^8-5)/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) = (1/2)*(4*_a^8+4*_a^3*_b(_a)-5)/_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: 0.156 (sec). Leaf size: 31
dsolve(diff(y(x),x) = 1/2*(1+2*(4*x^2*y(x)+1)^(1/2)*x^4)/x^3,y(x), singsol=all)
\[ \frac {2 x^{4}+3 c_{1} x -3 \sqrt {4 x^{2} y \left (x \right )+1}}{3 x} = 0 \]
✓ Solution by Mathematica
Time used: 0.331 (sec). Leaf size: 33
DSolve[y'[x] == (1/2 + x^4*Sqrt[1 + 4*x^2*y[x]])/x^3,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {x^6}{9}-\frac {2 c_1 x^3}{3}-\frac {1}{4 x^2}+c_1{}^2 \]