2.253 problem 829

Internal problem ID [9808]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 829
Date solved : Thursday, October 17, 2024 at 10:10:51 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]

Solve

\begin{align*} y^{\prime }&=\frac {1+2 \sqrt {4 x^{2} y+1}\, x^{3}+2 x^{5} \sqrt {4 x^{2} y+1}+2 x^{6} \sqrt {4 x^{2} y+1}}{2 x^{3}} \end{align*}

Unknown ode type.

2.253.1 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=\frac {1+2 \sqrt {1+4 y \left (x \right ) x^{2}}\, x^{3}+2 x^{5} \sqrt {1+4 y \left (x \right ) x^{2}}+2 x^{6} \sqrt {1+4 y \left (x \right ) x^{2}}}{2 x^{3}} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=\frac {1+2 \sqrt {1+4 y \left (x \right ) x^{2}}\, x^{3}+2 x^{5} \sqrt {1+4 y \left (x \right ) x^{2}}+2 x^{6} \sqrt {1+4 y \left (x \right ) x^{2}}}{2 x^{3}} \end {array} \]

2.253.2 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)-(4*x^3+3*x^2+1)*(diff(y(x), x))/(x*(x^3+x^2+1))-(1/2)*(4*x^15+12*x^14+12*x^ 
   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^15+12*_a^14+12*_a^13+16*_a^12+24*_a^11+12*_a^10+12*_a^9+12*_a^8+ 
      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`
 

2.253.3 Maple dsolve solution

Solving time : 0.201 (sec)
Leaf size : 41

dsolve(diff(y(x),x) = 1/2*(1+2*(4*x^2*y(x)+1)^(1/2)*x^3+2*x^5*(4*x^2*y(x)+1)^(1/2)+2*x^6*(4*x^2*y(x)+1)^(1/2))/x^3, 
       y(x),singsol=all)
 

\[ \frac {4 x^{6}+5 x^{5}+10 x^{3}+10 c_{1} x -10 \sqrt {4 x^{2} y+1}}{10 x} = 0 \]

2.253.4 Mathematica DSolve solution

Solving time : 0.601 (sec)
Leaf size : 81

DSolve[{D[y[x],x] == (1/2 + x^3*Sqrt[1 + 4*x^2*y[x]] + x^5*Sqrt[1 + 4*x^2*y[x]] + x^6*Sqrt[1 + 4*x^2*y[x]])/x^3,{}}, 
       y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {x^{10}}{25}+\frac {x^9}{10}+\frac {x^8}{16}+\frac {x^7}{5}+\frac {x^6}{4}-\frac {2 c_1 x^5}{5}-\frac {1}{4} (-1+2 c_1) x^4-\frac {1}{4 x^2}-c_1 x^2+c_1{}^2 \]