2.72 problem 648

Internal problem ID [9632]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 648
Date solved : Friday, October 11, 2024 at 11:18:31 AM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(x)]`]]

Solve

\begin{align*} y^{\prime }&=-\frac {x^{3} \left (\sqrt {a}\, x +\sqrt {a}-2 \sqrt {a \,x^{4}+8 y}\right ) \sqrt {a}}{2 \left (x +1\right )} \end{align*}

Unknown ode type.

2.72.1 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=-\frac {x^{3} \left (\sqrt {a}\, x +\sqrt {a}-2 \sqrt {a \,x^{4}+8 y \left (x \right )}\right ) \sqrt {a}}{2 \left (x +1\right )} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=-\frac {x^{3} \left (\sqrt {a}\, x +\sqrt {a}-2 \sqrt {a \,x^{4}+8 y \left (x \right )}\right ) \sqrt {a}}{2 \left (x +1\right )} \end {array} \]

2.72.2 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)*(8*x^3-x-1)*a*x^3/(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)*(8*_a^7*a-_a^5*a-a*_a^4+4*_b(_a)*_a^2+10*_b(_a)*_a+6*_b(_a))/((_a+1)^2 
      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 
--- Trying Lie symmetry methods, 1st order --- 
`, `-> Computing symmetries using: way = 3 
`, `-> Computing symmetries using: way = 5`[0, (a*x^4+8*y)^(1/2)]
 

2.72.3 Maple dsolve solution

Solving time : 0.073 (sec)
Leaf size : 49

dsolve(diff(y(x),x) = -1/2*x^3*(a^(1/2)*x+a^(1/2)-2*(x^4*a+8*y(x))^(1/2))*a^(1/2)/(x+1), 
       y(x),singsol=all)
 

\[ -\sqrt {x^{4} a +8 y}-4 \sqrt {a}\, \ln \left (x +1\right )+\frac {2 \left (2 x^{3}-3 x^{2}+6 x \right ) \sqrt {a}}{3}-c_{1} = 0 \]

2.72.4 Mathematica DSolve solution

Solving time : 0.78 (sec)
Leaf size : 96

DSolve[{D[y[x],x] == -1/2*(Sqrt[a]*x^3*(Sqrt[a] + Sqrt[a]*x - 2*Sqrt[a*x^4 + 8*y[x]]))/(1 + x),{}}, 
       y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {1}{72} a \left (16 x^6-48 x^5+123 x^4-96 c_1 x^3+72 (-1+2 c_1) x^2-48 \left (2 x^3-3 x^2+6 x+9-6 c_1\right ) \log (x+1)+144 \log ^2(x+1)-144 (-3+2 c_1) x+36 (3-2 c_1){}^2\right ) \]