2.118 problem 694

Internal problem ID [9678]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 694
Date solved : Friday, October 11, 2024 at 11:23:06 AM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]

Solve

\begin{align*} y^{\prime }&=\frac {x +1+2 \sqrt {4 x^{2} y+1}\, x^{3}}{2 x^{3} \left (x +1\right )} \end{align*}

Unknown ode type.

2.118.1 Maple step by step solution

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

2.118.2 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)-(diff(y(x), x))/(x*(x+1))-(1/2)*(4*x^6-3*x^2-7*x-4)/(x^4*(x+1)^2), y(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^6+2*_b(_a)*_a^4+2*_b(_a)*_a^3-3*_a^2-7*_a-4)/(_a^4*(_a+1)^2), _b 
      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`
 

2.118.3 Maple dsolve solution

Solving time : 0.180 (sec)
Leaf size : 36

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

\[ \frac {-2 \ln \left (x +1\right ) x +c_{1} x +2 x^{2}-\sqrt {4 x^{2} y+1}}{x} = 0 \]

2.118.4 Mathematica DSolve solution

Solving time : 1.167 (sec)
Leaf size : 50

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

\[ y(x)\to x^2-\frac {1}{4 x^2}+\frac {1}{4} \log ^2\left ((x+1)^2\right )-2 c_1 x+(-x+c_1) \log \left ((x+1)^2\right )+c_1{}^2 \]