Internal problem ID [9181]
Internal file name [OUTPUT/8116_Monday_June_06_2022_01_48_32_AM_70744539/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 847.
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.
\[ \boxed {y^{\prime }-\sqrt {x^{2}+2 x +1-4 y}-x^{2} \sqrt {x^{2}+2 x +1-4 y}-\sqrt {x^{2}+2 x +1-4 y}\, x^{3}=\frac {x}{2}+\frac {1}{2}} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }-\sqrt {x^{2}+2 x +1-4 y}-x^{2} \sqrt {x^{2}+2 x +1-4 y}-\sqrt {x^{2}+2 x +1-4 y}\, x^{3}=\frac {x}{2}+\frac {1}{2} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {x}{2}+\frac {1}{2}+\sqrt {x^{2}+2 x +1-4 y}+x^{2} \sqrt {x^{2}+2 x +1-4 y}+\sqrt {x^{2}+2 x +1-4 y}\, x^{3} \end {array} \]
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: 0.188 (sec). Leaf size: 34
dsolve(diff(y(x),x) = 1/2*x+1/2+(x^2+2*x+1-4*y(x))^(1/2)+x^2*(x^2+2*x+1-4*y(x))^(1/2)+x^3*(x^2+2*x+1-4*y(x))^(1/2),y(x), singsol=all)
\[ c_{1} -\frac {x^{4}}{2}-\frac {2 x^{3}}{3}-2 x -\sqrt {x^{2}+2 x +1-4 y \left (x \right )} = 0 \]
✓ Solution by Mathematica
Time used: 0.596 (sec). Leaf size: 69
DSolve[y'[x] == 1/2 + x/2 + Sqrt[1 + 2*x + x^2 - 4*y[x]] + x^2*Sqrt[1 + 2*x + x^2 - 4*y[x]] + x^3*Sqrt[1 + 2*x + x^2 - 4*y[x]],y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {1}{144} \left (-9 x^8-24 x^7-16 x^6-72 x^5+24 (-4+3 c_1) x^4+96 c_1 x^3-108 x^2+72 (1+4 c_1) x+36-144 c_1{}^2\right ) \]