Simultaneous Differential Equations

Simultaneous Differential Equations:

Let us suppose that x and y are functions of an independent variable ‘t’ connected by a system of first order equation with D = d/dt

By solving a system of linear algebraic equations in cancelling either of the dependent variables (x or y) operating (1) with g1 (D) and (2) with f1 (D), x cancels out by subtraction. We obtain a second order differential equation in y. Which can be solved x can be obtained independently by cancelling y or by substituting the obtained y (t) in a suitable equation.

Problems on Simultaneous Differential Equation:

1. Solve

Solution: Taking D = d/dt, we have the system of equations:

By considering dy/dt - 2x - 5y = 0 we get;

Equation (3) and (4) represent the complete solution of the given system of the equations;

2. Solve given x(0) = 8 and y(0) = 3

Solutions: Taking D = d/dt we have the system of the equations;

Hence y = -2e^at 5e^-t