Internal problem ID [8999]
Internal file name [OUTPUT/7934_Monday_June_06_2022_12_57_38_AM_5041517/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 665.
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)]`]]
Unable to solve or complete the solution.
\[ \boxed {y^{\prime }+\frac {\left (\sqrt {a}\, x^{4}+\sqrt {a}\, x^{3}-2 \sqrt {a \,x^{4}+8 y}\right ) \sqrt {a}}{2 x +2}=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & -a \,x^{4}-a \,x^{3}+2 \sqrt {a}\, \sqrt {a \,x^{4}+8 y}-2 y^{\prime } x -2 y^{\prime }=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {a \,x^{4}+a \,x^{3}-2 \sqrt {a}\, \sqrt {a \,x^{4}+8 y}}{-2 x -2} \end {array} \]
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)+(diff(y(x), x))/(x+1)+(1/2)*a*(4*x^4+7*x^3+3*x^2-8)/(x+1)^2, y(x)` *** Su 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*_a^4+7*_a^3*a+3*_a^2*a+2*_b(_a)*_a+2*_b(_a)-8*a)/(_a+1)^2, _b(_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 --- Trying Lie symmetry methods, 1st order --- `, `-> Computing symmetries using: way = 3 `, `-> Computing symmetries using: way = 5`[0, (a*x^4+8*y)^(1/2)]
✓ Solution by Maple
Time used: 0.078 (sec). Leaf size: 30
dsolve(diff(y(x),x) = -1/2*(a^(1/2)*x^4+a^(1/2)*x^3-2*(a*x^4+8*y(x))^(1/2))*a^(1/2)/(x+1),y(x), singsol=all)
\[ 4 \sqrt {a}\, \ln \left (x +1\right )-\sqrt {x^{4} a +8 y \left (x \right )}-c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 0.686 (sec). Leaf size: 39
DSolve[y'[x] == -1/2*(Sqrt[a]*(Sqrt[a]*x^3 + Sqrt[a]*x^4 - 2*Sqrt[a*x^4 + 8*y[x]]))/(1 + x),y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to -\frac {a x^4}{8}+2 a \log ^2(x+1)-4 a c_1 \log (x+1)+2 a c_1{}^2 \]