2.102 problem 678

Internal problem ID [9657]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 678
Date solved : Thursday, October 17, 2024 at 08:56:21 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(x)]`]]

Solve

\begin{align*} y^{\prime }&=\frac {x^{2} \left (x +1+2 x \sqrt {x^{3}-6 y}\right )}{2 x +2} \end{align*}

Unknown ode type.

2.102.1 Maple step by step solution

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

2.102.2 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)-(2*x+3)*(diff(y(x), x))/((x+1)*x)+(1/2)*x*(6*x^5+x+1)/(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)*(-6*_a^7+4*_b(_a)*_a^2-_a^3+10*_b(_a)*_a-_a^2+6*_b(_a))/((_a+1)^2*_a), 
      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`
 

2.102.3 Maple dsolve solution

Solving time : 0.103 (sec)
Leaf size : 37

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

\[ c_{1} -x^{3}+\frac {3 x^{2}}{2}-3 x +3 \ln \left (x +1\right )-\frac {1}{2}-\sqrt {x^{3}-6 y} = 0 \]

2.102.4 Mathematica DSolve solution

Solving time : 4.241 (sec)
Leaf size : 99

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

\[ y(x)\to \frac {1}{24} \left (-4 x^6+12 x^5-33 x^4+4 (-1+6 c_1) x^3-6 (-5+6 c_1) x^2+12 \left (2 x^3-3 x^2+6 x+11-6 c_1\right ) \log (x+1)-36 \log ^2(x+1)+12 (-11+6 c_1) x-(11-6 c_1){}^2\right ) \]