Internal problem ID [9147]
Internal file name [OUTPUT/8082_Monday_June_06_2022_01_41_48_AM_10636525/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 813.
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.
\[ \boxed {y^{\prime }-\frac {\left (-x^{3} \sqrt {a}+2 \sqrt {a \,x^{4}+8 y}+2 x^{2} \sqrt {a \,x^{4}+8 y}+2 x^{3} \sqrt {a \,x^{4}+8 y}\right ) \sqrt {a}}{2}=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }-\frac {\left (-x^{3} \sqrt {a}+2 \sqrt {a \,x^{4}+8 y}+2 x^{2} \sqrt {a \,x^{4}+8 y}+2 x^{3} \sqrt {a \,x^{4}+8 y}\right ) \sqrt {a}}{2}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {\left (-x^{3} \sqrt {a}+2 \sqrt {a \,x^{4}+8 y}+2 x^{2} \sqrt {a \,x^{4}+8 y}+2 x^{3} \sqrt {a \,x^{4}+8 y}\right ) \sqrt {a}}{2} \end {array} \]
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)-x*(3*x+2)*(diff(y(x), x))/(x^3+x^2+1)-(1/2)*(8*x^9+24*x^8+24*x^7+32*x^6+48* 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*_a^9+24*_a^8*a+24*_a^7*a+32*_a^6*a+48*_a^5*a+23*a*_a^4+24*_a^3*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: 40
dsolve(diff(y(x),x) = 1/2*(-a^(1/2)*x^3+2*(a*x^4+8*y(x))^(1/2)+2*x^2*(a*x^4+8*y(x))^(1/2)+2*x^3*(a*x^4+8*y(x))^(1/2))*a^(1/2),y(x), singsol=all)
\[ \frac {\sqrt {x^{4} a +8 y \left (x \right )}}{4}+\frac {\left (-3 x^{4}-4 x^{3}-12 x \right ) \sqrt {a}}{12}-c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 0.872 (sec). Leaf size: 64
DSolve[y'[x] == (Sqrt[a]*(-(Sqrt[a]*x^3) + 2*Sqrt[a*x^4 + 8*y[x]] + 2*x^2*Sqrt[a*x^4 + 8*y[x]] + 2*x^3*Sqrt[a*x^4 + 8*y[x]]))/2,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {1}{72} a \left (9 x^8+24 x^7+16 x^6+72 x^5+(87-72 c_1) x^4-96 c_1 x^3+144 x^2-288 c_1 x+144 c_1{}^2\right ) \]