Internal problem ID [9191]
Internal file name [OUTPUT/8126_Monday_June_06_2022_01_50_47_AM_53221421/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 857.
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.
\[ \boxed {y^{\prime }-\sqrt {x^{2}-2 x +1+8 y}-x^{2} \sqrt {x^{2}-2 x +1+8 y}-x^{3} \sqrt {x^{2}-2 x +1+8 y}=-\frac {x}{4}+\frac {1}{4}} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }-\sqrt {x^{2}-2 x +1+8 y}-x^{2} \sqrt {x^{2}-2 x +1+8 y}-x^{3} \sqrt {x^{2}-2 x +1+8 y}=-\frac {x}{4}+\frac {1}{4} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-\frac {x}{4}+\frac {1}{4}+\sqrt {x^{2}-2 x +1+8 y}+x^{2} \sqrt {x^{2}-2 x +1+8 y}+x^{3} \sqrt {x^{2}-2 x +1+8 y} \end {array} \]
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)-(1/4)*(16*x^9+48*x^8+48*x^7+64*x^6+96 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) = (1/4)*(16*_a^9+48*_a^8+48*_a^7+64*_a^6+96*_a^5+48*_a^4+12*_b(_a)*_a^2+50*_a^ 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: 0.156 (sec). Leaf size: 32
dsolve(diff(y(x),x) = -1/4*x+1/4+(x^2-2*x+1+8*y(x))^(1/2)+x^2*(x^2-2*x+1+8*y(x))^(1/2)+x^3*(x^2-2*x+1+8*y(x))^(1/2),y(x), singsol=all)
\[ c_{1} +x^{4}+\frac {4 x^{3}}{3}+4 x -\sqrt {x^{2}-2 x +1+8 y \left (x \right )} = 0 \]
✓ Solution by Mathematica
Time used: 0.798 (sec). Leaf size: 77
DSolve[y'[x] == 1/4 - x/4 + Sqrt[1 - 2*x + x^2 + 8*y[x]] + x^2*Sqrt[1 - 2*x + x^2 + 8*y[x]] + x^3*Sqrt[1 - 2*x + x^2 + 8*y[x]],y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {x^8}{8}+\frac {x^7}{3}+\frac {2 x^6}{9}+x^5+\left (\frac {4}{3}-c_1\right ) x^4-\frac {4 c_1 x^3}{3}+\frac {15 x^2}{8}+\left (\frac {1}{4}-4 c_1\right ) x-\frac {1}{8}+2 c_1{}^2 \]