Internal problem ID [9146]
Internal file name [OUTPUT/8081_Monday_June_06_2022_01_41_35_AM_56264210/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 812.
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.
\[ \boxed {y^{\prime }-\sqrt {x^{3}-6 y}-x^{2} \sqrt {x^{3}-6 y}-x^{3} \sqrt {x^{3}-6 y}=\frac {x^{2}}{2}} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }-\sqrt {x^{3}-6 y}-x^{2} \sqrt {x^{3}-6 y}-x^{3} \sqrt {x^{3}-6 y}=\frac {x^{2}}{2} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {x^{2}}{2}+\sqrt {x^{3}-6 y}+x^{2} \sqrt {x^{3}-6 y}+x^{3} \sqrt {x^{3}-6 y} \end {array} \]
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)*(6*x^9+18*x^8+18*x^7+24*x^6+36* 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^9-18*_a^8-18*_a^7-24*_a^6-36*_a^5-19*_a^4+6*_b(_a)*_a^2-18*_a^3 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`
✓ Solution by Maple
Time used: 0.187 (sec). Leaf size: 30
dsolve(diff(y(x),x) = 1/2*x^2+(x^3-6*y(x))^(1/2)+x^2*(x^3-6*y(x))^(1/2)+x^3*(x^3-6*y(x))^(1/2),y(x), singsol=all)
\[ c_{1} -\frac {3 x^{4}}{4}-x^{3}-3 x -\sqrt {x^{3}-6 y \left (x \right )} = 0 \]
✓ Solution by Mathematica
Time used: 0.618 (sec). Leaf size: 76
DSolve[y'[x] == x^2/2 + Sqrt[x^3 - 6*y[x]] + x^2*Sqrt[x^3 - 6*y[x]] + x^3*Sqrt[x^3 - 6*y[x]],y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to -\frac {3 x^8}{32}-\frac {x^7}{4}-\frac {x^6}{6}-\frac {3 x^5}{4}+\left (-1+\frac {3 c_1}{4}\right ) x^4+\left (\frac {1}{6}+c_1\right ) x^3-\frac {3 x^2}{2}+3 c_1 x-\frac {3 c_1{}^2}{2} \]