Internal problem ID [9199]
Internal file name [OUTPUT/8135_Monday_June_06_2022_01_52_47_AM_6794827/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 866.
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.
\[ \boxed {y^{\prime }-\sqrt {a^{2}+2 x a +x^{2}+4 y}-x^{2} \sqrt {a^{2}+2 x a +x^{2}+4 y}-\sqrt {a^{2}+2 x a +x^{2}+4 y}\, x^{3}=-\frac {x}{2}-\frac {a}{2}} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }-\sqrt {a^{2}+2 x a +x^{2}+4 y}-x^{2} \sqrt {a^{2}+2 x a +x^{2}+4 y}-\sqrt {a^{2}+2 x a +x^{2}+4 y}\, x^{3}=-\frac {x}{2}-\frac {a}{2} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-\frac {x}{2}-\frac {a}{2}+\sqrt {a^{2}+2 x a +x^{2}+4 y}+x^{2} \sqrt {a^{2}+2 x a +x^{2}+4 y}+\sqrt {a^{2}+2 x a +x^{2}+4 y}\, x^{3} \end {array} \]
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)-(1/2)*(4*x^9+12*x^8+12*x^7+16*x^6+24* 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^9+12*_a^8+12*_a^7+16*_a^6+24*_a^5+12*_a^4+6*_b(_a)*_a^2+14*_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: 37
dsolve(diff(y(x),x) = -1/2*x-1/2*a+(x^2+2*a*x+a^2+4*y(x))^(1/2)+x^2*(x^2+2*a*x+a^2+4*y(x))^(1/2)+x^3*(x^2+2*a*x+a^2+4*y(x))^(1/2),y(x), singsol=all)
\[ c_{1} +\frac {x^{4}}{2}+\frac {2 x^{3}}{3}+2 x -\sqrt {x^{2}+2 a x +a^{2}+4 y \left (x \right )} = 0 \]
✓ Solution by Mathematica
Time used: 0.784 (sec). Leaf size: 85
DSolve[y'[x] == -1/2*a - x/2 + Sqrt[a^2 + 2*a*x + x^2 + 4*y[x]] + x^2*Sqrt[a^2 + 2*a*x + x^2 + 4*y[x]] + x^3*Sqrt[a^2 + 2*a*x + x^2 + 4*y[x]],y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to -\frac {a^2}{4}-\frac {a x}{2}+\frac {x^8}{16}+\frac {x^7}{6}+\frac {x^6}{9}+\frac {x^5}{2}-\frac {1}{6} (-4+3 c_1) x^4-\frac {2 c_1 x^3}{3}+\frac {3 x^2}{4}-2 c_1 x+c_1{}^2 \]