Internal problem ID [13056]
Internal file name [OUTPUT/11709_Wednesday_November_08_2023_03_29_05_AM_81720901/index.tex]
Book: DIFFERENTIAL EQUATIONS by Paul Blanchard, Robert L. Devaney, Glen R. Hall. 4th
edition. Brooks/Cole. Boston, USA. 2012
Section: Chapter 1. First-Order Differential Equations. Review Exercises for chapter 1. page
136
Problem number: 44.
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.
\[ \boxed {y^{\prime }-\left (y-1\right ) \left (y-2\right ) \left (y-{\mathrm e}^{\frac {t}{2}}\right )=0} \]
This is Abel first kind ODE, it has the form \[ y^{\prime }= f_0(t)+f_1(t) y +f_2(t)y^{2}+f_3(t)y^{3} \] Comparing the above to given ODE which is \begin {align*} y^{\prime }&=y^{3}+\left (-3-{\mathrm e}^{\frac {t}{2}}\right ) y^{2}+\left (3 \,{\mathrm e}^{\frac {t}{2}}+2\right ) y-2 \,{\mathrm e}^{\frac {t}{2}}\tag {1} \end {align*}
Therefore \begin {align*} f_0(t) &= -2 \,{\mathrm e}^{\frac {t}{2}}\\ f_1(t) &= 3 \,{\mathrm e}^{\frac {t}{2}}+2\\ f_2(t) &= -3-{\mathrm e}^{\frac {t}{2}}\\ f_3(t) &= 1 \end {align*}
Since \(f_2(t)=-3-{\mathrm e}^{\frac {t}{2}}\) is not zero, then the first step is to apply the following transformation to remove \(f_2\). Let \(y = u(t) - \frac {f_2}{3 f_3}\) or \begin {align*} y &= u(t) - \left ( \frac {-3-{\mathrm e}^{\frac {t}{2}}}{3} \right ) \\ &= u \left (t \right )+1+\frac {{\mathrm e}^{\frac {t}{2}}}{3} \end {align*}
The above transformation applied to (1) gives a new ODE as \begin {align*} u^{\prime }\left (t \right ) = -\frac {{\mathrm e}^{\frac {t}{2}}}{2}+u \left (t \right )^{3}-u \left (t \right )+u \left (t \right ) {\mathrm e}^{\frac {t}{2}}-\frac {u \left (t \right ) {\mathrm e}^{t}}{3}+\frac {{\mathrm e}^{t}}{3}-\frac {2 \,{\mathrm e}^{\frac {3 t}{2}}}{27}\tag {2} \end {align*}
This is Abel first kind ODE, it has the form \[ u^{\prime }\left (t \right )= f_0(t)+f_1(t) u \left (t \right ) +f_2(t)u \left (t \right )^{2}+f_3(t)u \left (t \right )^{3} \] Comparing the above to given ODE which is \begin {align*} u^{\prime }\left (t \right )&=u \left (t \right )^{3}+\left (-1+{\mathrm e}^{\frac {t}{2}}-\frac {{\mathrm e}^{t}}{3}\right ) u \left (t \right )-\frac {{\mathrm e}^{\frac {t}{2}}}{2}+\frac {{\mathrm e}^{t}}{3}-\frac {2 \,{\mathrm e}^{\frac {3 t}{2}}}{27}\tag {1} \end {align*}
Therefore \begin {align*} f_0(t) &= -\frac {{\mathrm e}^{\frac {t}{2}}}{2}+\frac {{\mathrm e}^{t}}{3}-\frac {2 \,{\mathrm e}^{\frac {3 t}{2}}}{27}\\ f_1(t) &= -1+{\mathrm e}^{\frac {t}{2}}-\frac {{\mathrm e}^{t}}{3}\\ f_2(t) &= 0\\ f_3(t) &= 1 \end {align*}
Since \(f_2(t)=0\) then we check the Abel invariant to see if it depends on \(t\) 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*} -\frac {\left (\frac {{\mathrm e}^{\frac {t}{2}}}{4}-\frac {{\mathrm e}^{t}}{3}+\frac {{\mathrm e}^{\frac {3 t}{2}}}{9}+3 \left (-\frac {{\mathrm e}^{\frac {t}{2}}}{2}+\frac {{\mathrm e}^{t}}{3}-\frac {2 \,{\mathrm e}^{\frac {3 t}{2}}}{27}\right ) \left (-1+{\mathrm e}^{\frac {t}{2}}-\frac {{\mathrm e}^{t}}{3}\right )\right )^{3}}{27 \left (-\frac {{\mathrm e}^{\frac {t}{2}}}{2}+\frac {{\mathrm e}^{t}}{3}-\frac {2 \,{\mathrm e}^{\frac {3 t}{2}}}{27}\right )^{5}} \end {align*}
Since the Abel invariant depends on \(t\) then unable to solve this ode at this time.
Unable to complete the solution now.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }-\left (y-1\right ) \left (y-2\right ) \left (y-{\mathrm e}^{\frac {t}{2}}\right )=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\left (y-1\right ) \left (y-2\right ) \left (y-{\mathrm e}^{\frac {t}{2}}\right ) \end {array} \]
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 -> Calling odsolve with the ODE`, diff(y(x), x) = -1/(exp(y(x)+x-2*exp((1/2)*x)-1)-1), y(x), explicit` *** Sublevel 2 *** 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 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 = 5 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)] -> 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 -> Calling odsolve with the ODE`, diff(y(x), x) = -exp(x+y(x)-2*exp((1/2)*y(x))-1)+1, y(x), implicit` *** Sublevel 2 *** 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 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 = 5 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)] -> 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 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)-3, 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)+(1/2)*(y(x)*exp((1/2)*x)-4)/exp((1/2)*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)*(2*x-3)/((x-1)*(x-2)), 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) = y(x)*(3*x^2-6*x+2)/((x-1)*(x-2)*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*K[1], 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)+(1/2)*(y(x)*exp((1/2)*x)-4*K[1])/exp((1/2)*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*K[1]*exp((1/2)*x), y(x)` *** Sublevel 2 *** Methods for first order ODEs: --- Trying classification methods --- trying a quadrature <- quadrature successful -> Calling odsolve with the ODE`, diff(y(x), x)+(1/2)*y(x)-2*K[1], 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) = -(1/2)*y(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)-y(x)*(3*x^2-6*x+2)/((x-1)*(x-2)*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)-(1/2)*(K[1]*x^2+4*y(x)*x-3*K[1]*x-6*y(x)+2*K[1])/((x-1)*(x-2)), y(x)` *** Su 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)-y(x)*(2*x-3)/((x-1)*(x-2)), 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(t),t)=(y(t)-1)*(y(t)-2)*(y(t)-exp(t/2)),y(t), singsol=all)
\[ \text {No solution found} \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[y'[t]==(y[t]-1)*(y[t]-2)*(y[t]-Exp[t/2]),y[t],t,IncludeSingularSolutions -> True]
Timed out