The rate at which a chemical process occurs is usually best described as a differential equation.
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In its normal form, you do not solve differential equation for x, but for a function of x, usually denoted by y = f(x).
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Differential equation is defined in the domain except at few points (may be consider the time domain ti ) may be (finite or countable) in the domain and a function or difference equation is defined at each ti in the domain. So, differential equation with the impulsive effects we call it as impulsive differential equation (IDE). The solutions of the differential equation is continuous in the domain. But the solutions of the IDE are piecewise continuous in the domain. This is due to the nature of impulsive system. Generally IDE have first order discontinuity. There are so many applications for IDE in practical life.
PDE stands for Partial Differential Equation
Many real world problems can be represented by first order differential equation. Some applications of differential equation are radio-active decay and carbon dating, population growth and decay, warming/cooling law and draining a tank.