Internal problem ID [8921]
Internal file name [OUTPUT/7856_Monday_June_06_2022_12_47_40_AM_49797003/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 587.
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.
\[ \boxed {y^{\prime }-\frac {\left (x^{\frac {3}{2}}+2 F \left (y-\frac {x^{3}}{6}\right )\right ) \sqrt {x}}{2}=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }-\frac {\left (x^{\frac {3}{2}}+2 F \left (y-\frac {x^{3}}{6}\right )\right ) \sqrt {x}}{2}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {\left (x^{\frac {3}{2}}+2 F \left (y-\frac {x^{3}}{6}\right )\right ) \sqrt {x}}{2} \end {array} \]
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 --- Trying Lie symmetry methods, 1st order --- `, `-> Computing symmetries using: way = 3 `, `-> Computing symmetries using: way = 4 `, `-> Computing symmetries using: way = 5`[0, F(y-1/6*x^3)]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 29
dsolve(diff(y(x),x) = 1/2*(x^(3/2)+2*F(y(x)-1/6*x^3))*x^(1/2),y(x), singsol=all)
\[ \int _{\textit {\_b}}^{y \left (x \right )}\frac {1}{F \left (\textit {\_a} -\frac {x^{3}}{6}\right )}d \textit {\_a} -\frac {2 x^{\frac {3}{2}}}{3}-c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 0.274 (sec). Leaf size: 123
DSolve[y'[x] == (Sqrt[x]*(x^(3/2) + 2*F[-1/6*x^3 + y[x]]))/2,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\int _1^{y(x)}-\frac {F\left (K[2]-\frac {x^3}{6}\right ) \int _1^x-\frac {K[1]^2 F'\left (K[2]-\frac {K[1]^3}{6}\right )}{2 F\left (K[2]-\frac {K[1]^3}{6}\right )^2}dK[1]+1}{F\left (K[2]-\frac {x^3}{6}\right )}dK[2]+\int _1^x\left (\frac {K[1]^2}{2 F\left (y(x)-\frac {K[1]^3}{6}\right )}+\sqrt {K[1]}\right )dK[1]=c_1,y(x)\right ] \]