Internal problem ID [9153]
Internal file name [OUTPUT/8088_Monday_June_06_2022_01_43_15_AM_84746700/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 819.
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.
\[ \boxed {y^{\prime }-\sqrt {x^{2}+3 y}-x^{2} \sqrt {x^{2}+3 y}-\sqrt {x^{2}+3 y}\, x^{3}=-\frac {2 x}{3}} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }-\sqrt {x^{2}+3 y}-x^{2} \sqrt {x^{2}+3 y}-\sqrt {x^{2}+3 y}\, x^{3}=-\frac {2 x}{3} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-\frac {2 x}{3}+\sqrt {x^{2}+3 y}+x^{2} \sqrt {x^{2}+3 y}+\sqrt {x^{2}+3 y}\, x^{3} \end {array} \]
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)-(1/6)*(9*x^9+27*x^8+27*x^7+36*x^6+54* 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) = (1/6)*(9*_a^9+27*_a^8+27*_a^7+36*_a^6+54*_a^5+27*_a^4+18*_b(_a)*_a^2+35*_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: 0.172 (sec). Leaf size: 30
dsolve(diff(y(x),x) = -2/3*x+(x^2+3*y(x))^(1/2)+x^2*(x^2+3*y(x))^(1/2)+x^3*(x^2+3*y(x))^(1/2),y(x), singsol=all)
\[ c_{1} +\frac {3 x^{4}}{8}+\frac {x^{3}}{2}+\frac {3 x}{2}-\sqrt {x^{2}+3 y \left (x \right )} = 0 \]
✓ Solution by Mathematica
Time used: 0.471 (sec). Leaf size: 63
DSolve[y'[x] == (-2*x)/3 + Sqrt[x^2 + 3*y[x]] + x^2*Sqrt[x^2 + 3*y[x]] + x^3*Sqrt[x^2 + 3*y[x]],y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {1}{192} \left (9 x^8+24 x^7+16 x^6+72 x^5+(96-72 c_1) x^4-96 c_1 x^3+80 x^2-288 c_1 x+144 c_1{}^2\right ) \]