Consider the system of Ordinary differential equations $\frac{dx(t)}{dt}=-\alpha_1-\beta_1x(t)-\beta_3y(t)\tag{1}$ $\frac{dy(t)}{dt}=-\alpha_2-\beta_2y(t)-\beta_3x(t)\tag{2}$ With initial conditions $x(0)=p_1,y(0)=p_2$

Solution:Let $\frac{d}{dt}=D$

Then from $(1)$,We have

$(D+\beta_1)x(t)+\alpha _1+\beta_3y(t)=0\tag{3}$

From $(2)$,we have

$(D+\beta_2)y(t)+\alpha _2+\beta_3x(t)=0\tag{4}$

Eliminating $y(t)$ with the help of $(3)$ and $(4)$.Multiplying equation $(3)$ by $(D+b_2)$ and subtracting $eq^n(4)\times\beta_3$,we get

$(D^2+(\beta_1+\beta_2 )D+\beta_1 \beta_2-\beta_3^2)x(t)+\beta_1\alpha_1-\alpha_2\beta_3=0\tag{5}$.

The complementary function of the above $eq^n(5)$ is

$x(t)=c_1 e^{r_1}+c_2e^{r^2},$where $r_1=\frac{-(\beta_1+\beta_2)+{\sqrt{(\beta_1-\beta_2-2\beta_3)(\beta_1-\beta_2+2\beta_3)}}}{2} $ and $r_2=\frac{-(\beta_1+\beta_2)-{\sqrt{(\beta_1-\beta_2-2\beta_3)(\beta_1-\beta_2+2\beta_3)}}}{2}$

From here , i'm unable to proceed further.I've invested a great amount of time on this problem.I would be nice if some one could come up with the solution.

The answer of this problem is

$x(t)=(\beta_1p_1-\beta_3p_2-\alpha_1)t+p_1$, $y(t)=(\beta_1p_2-\beta_3p_1-\alpha_2)t+p_2$

  • 1
    $\begingroup$ What you wrote is not the solution of the IVP. Do you know how to solve a linear non-homogeneous ordinary differential equation with constant coefficients? What is the characteristic polynomial of a linear homogeneous ODE with constant coefficients? The method of undetermined coefficients? $\endgroup$ – user539887 Jun 29 '18 at 6:20

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