Internal problem ID [9010]
Internal file name [OUTPUT/7945_Monday_June_06_2022_12_59_24_AM_41104474/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 676.
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.
\[ \boxed {y^{\prime }-\frac {x +1+2 x^{6} \sqrt {1+4 y x^{2}}}{2 x^{3} \left (1+x \right )}=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & 2 x^{6} \sqrt {1+4 y x^{2}}-2 y^{\prime } x^{4}-2 y^{\prime } x^{3}+x +1=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {-x -1-2 x^{6} \sqrt {1+4 y x^{2}}}{-2 x^{4}-2 x^{3}} \end {array} \]
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)-(3*x+4)*(diff(y(x), x))/(x*(x+1))-(1/2)*(4*x^12-6*x^2-13*x-7)/((x+1)^2*x^4) 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^12+6*_b(_a)*_a^5+14*_b(_a)*_a^4+8*_a^3*_b(_a)-6*_a^2-13*_a-7)/(( 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: 0.266 (sec). Leaf size: 53
dsolve(diff(y(x),x) = 1/2*(x+1+2*x^6*(4*x^2*y(x)+1)^(1/2))/x^3/(x+1),y(x), singsol=all)
\[ \frac {3 x^{5}-4 x^{4}+6 x^{3}+12 \ln \left (x +1\right ) x +6 c_{1} x -12 x^{2}-6 \sqrt {4 x^{2} y \left (x \right )+1}}{6 x} = 0 \]
✓ Solution by Mathematica
Time used: 13.892 (sec). Leaf size: 83
DSolve[y'[x] == (1/2 + x/2 + x^6*Sqrt[1 + 4*x^2*y[x]])/(x^3*(1 + x)),y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {1}{4} \left (-\frac {1}{x^2}+\log ^2\left ((x+1)^2 \left (\cosh \left (-\frac {x^4}{2}+\frac {2 x^3}{3}-x^2+2 x+2 c_1\right )-\sinh \left (-\frac {x^4}{2}+\frac {2 x^3}{3}-x^2+2 x+2 c_1\right )\right )\right )\right ) \]