Internal problem ID [8384]
Internal file name [OUTPUT/7317_Sunday_June_05_2022_05_47_19_PM_46327166/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 47.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "abelFirstKind"
Maple gives the following as the ode type
[_Abel]
Unable to solve or complete the solution.
\[ \boxed {y^{\prime }-a \left (x^{n}-x \right ) y^{3}-y^{2}=0} \]
This is Abel first kind ODE, it has the form \[ y^{\prime }= f_0(x)+f_1(x) y +f_2(x)y^{2}+f_3(x)y^{3} \] Comparing the above to given ODE which is \begin {align*} y^{\prime }&=\left (x^{n} a -x a \right ) y^{3}+y^{2}\tag {1} \end {align*}
Therefore \begin {align*} f_0(x) &= 0\\ f_1(x) &= 0\\ f_2(x) &= 1\\ f_3(x) &= x^{n} a -x a \end {align*}
Since \(f_2(x)=1\) is not zero, then the first step is to apply the following transformation to remove \(f_2\). Let \(y = u(x) - \frac {f_2}{3 f_3}\) or \begin {align*} y &= u(x) - \left ( \frac {1}{3 x^{n} a -3 x a} \right ) \\ &= u \left (x \right )-\frac {1}{3 a \left (x^{n}-x \right )} \end {align*}
The above transformation applied to (1) gives a new ODE as \begin {align*} u^{\prime }\left (x \right ) = \frac {a \,x^{3 n} u \left (x \right )^{3}}{\left (x^{n}-x \right )^{2}}-\frac {3 a x \,x^{2 n} u \left (x \right )^{3}}{\left (x^{n}-x \right )^{2}}+\frac {3 a \,x^{2} x^{n} u \left (x \right )^{3}}{\left (x^{n}-x \right )^{2}}-\frac {a \,x^{3} u \left (x \right )^{3}}{\left (x^{n}-x \right )^{2}}-\frac {x^{n} u \left (x \right )}{3 a \left (x^{n}-x \right )^{2}}+\frac {x u \left (x \right )}{3 a \left (x^{n}-x \right )^{2}}-\frac {x^{n} n}{3 a x \left (x^{n}-x \right )^{2}}+\frac {1}{3 a \left (x^{n}-x \right )^{2}}+\frac {2}{27 a^{2} \left (x^{n}-x \right )^{2}}\tag {2} \end {align*}
This is Abel first kind ODE, it has the form \[ u^{\prime }\left (x \right )= f_0(x)+f_1(x) u \left (x \right ) +f_2(x)u \left (x \right )^{2}+f_3(x)u \left (x \right )^{3} \] Comparing the above to given ODE which is \begin {align*} u^{\prime }\left (x \right )&=\frac {\left (27 x^{3 n} a^{3} x -81 x^{2 n} a^{3} x^{2}+81 x^{n} a^{3} x^{3}-27 a^{3} x^{4}\right ) u \left (x \right )^{3}}{27 a^{2} x \left (x^{2 n}-2 x \,x^{n}+x^{2}\right )}+\frac {\left (-9 x a \,x^{n}+9 a \,x^{2}\right ) u \left (x \right )}{27 a^{2} x \left (x^{2 n}-2 x \,x^{n}+x^{2}\right )}+\frac {-9 x^{n} n a +9 x a +2 x}{27 a^{2} x \left (x^{2 n}-2 x \,x^{n}+x^{2}\right )}\tag {1} \end {align*}
Therefore \begin {align*} f_0(x) &= -\frac {x^{n} n}{3 a x \left (x^{2 n}-2 x \,x^{n}+x^{2}\right )}+\frac {1}{3 a \left (x^{2 n}-2 x \,x^{n}+x^{2}\right )}+\frac {2}{27 a^{2} \left (x^{2 n}-2 x \,x^{n}+x^{2}\right )}\\ f_1(x) &= -\frac {x^{n}}{3 a \left (x^{2 n}-2 x \,x^{n}+x^{2}\right )}+\frac {x}{3 a \left (x^{2 n}-2 x \,x^{n}+x^{2}\right )}\\ f_2(x) &= 0\\ f_3(x) &= \frac {a \,x^{3 n}}{x^{2 n}-2 x \,x^{n}+x^{2}}-\frac {3 x a \,x^{2 n}}{x^{2 n}-2 x \,x^{n}+x^{2}}+\frac {3 a \,x^{2} x^{n}}{x^{2 n}-2 x \,x^{n}+x^{2}}-\frac {a \,x^{3}}{x^{2 n}-2 x \,x^{n}+x^{2}} \end {align*}
Since \(f_2(x)=0\) then we check the Abel invariant to see if it depends on \(x\) or not. The Abel invariant is given by \begin {align*} -\frac {f_{1}^{3}}{f_{0}^{2} f_{3}} \end {align*}
Which when evaluating gives \begin {align*} \text {Expression too large to display} \end {align*}
Since the Abel invariant depends on \(x\) then unable to solve this ode at this time.
Unable to complete the solution now.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }-a \left (x^{n}-x \right ) y^{3}-y^{2}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=a \left (x^{n}-x \right ) y^{3}+y^{2} \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear trying Bernoulli trying separable trying inverse linear trying homogeneous types: trying Chini differential order: 1; looking for linear symmetries trying exact trying Abel Looking for potential symmetries Looking for potential symmetries Looking for potential symmetries trying inverse_Riccati trying an equivalence to an Abel ODE differential order: 1; trying a linearization to 2nd order --- trying a change of variables {x -> y(x), y(x) -> x} differential order: 1; trying a linearization to 2nd order 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 = 2 trying symmetry patterns for 1st order ODEs -> trying a symmetry pattern of the form [F(x)*G(y), 0] -> trying a symmetry pattern of the form [0, F(x)*G(y)] -> trying symmetry patterns of the forms [F(x),G(y)] and [G(y),F(x)] `, `-> Computing symmetries using: way = HINT -> Calling odsolve with the ODE`, diff(y(x), x)+y(x)*(x^n*n-x)/(x*(x^n-x)), y(x)` *** Sublevel 2 *** Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear <- 1st order linear successful `, `-> Computing symmetries using: way = HINT -> Calling odsolve with the ODE`, diff(y(x), x) = -y(x)*(x^n*n-x)/(x*(x^n-x)), y(x)` *** Sublevel 2 *** Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear <- 1st order linear successful -> Calling odsolve with the ODE`, diff(y(x), x)-(-a*x^2*K[1]+2*y(x))/x, y(x)` *** Sublevel 2 *** Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear <- 1st order linear successful -> Calling odsolve with the ODE`, diff(y(x), x)-3*y(x)/x, y(x)` *** Sublevel 2 *** Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear <- 1st order linear successful -> trying a symmetry pattern of the form [F(x),G(x)] -> trying a symmetry pattern of the form [F(y),G(y)] -> trying a symmetry pattern of the form [F(x)+G(y), 0] -> trying a symmetry pattern of the form [0, F(x)+G(y)] -> trying a symmetry pattern of the form [F(x),G(x)*y+H(x)] -> trying a symmetry pattern of conformal type`
✗ Solution by Maple
dsolve(diff(y(x),x) - a*(x^n - x)*y(x)^3 - y(x)^2=0,y(x), singsol=all)
\[ \text {No solution found} \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[y'[x] - a*(x^n - x)*y[x]^3 - y[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
Not solved