Introduction to Differential Equations

 What is a differential equation? An equation that involves one or more derivatives of an unknown function is called a differential equation. The order of the highest derivative included in a differential equation defines the order of this equation. Examples y ' = 3x , the order of the highest derivative is 1 (y ' ) so the order of this differential equation is 1. y '' + y' + y = 3x , the order of the highest derivative is 2 (y '' ) so the order of this differential equation is 2. -2 y ''' + y'' + y 4 = 3x , the order of the highest derivative is 3 (y ''' ) so the order of this differential equation is 3. y = f(x) is a solution of a differential equation if the equation is satisfied upon substitution of y and its derivatives into the differential equation. Example: Verify that y = C*e 4x + e 3x, where c is a constant, is a solution to the differential equation y ' - 4y = -e 3x y ' is given by y ' = 4C*e 4x + 3e 3x We now substitute y ' and y into the left side of the equation and simplify y ' - 4y = 4C*e 4x + 3e 3x - 4 (C*e 4x + e 3x) = 4C*e 4x + 3e 3x - 4C*e 4x - 4e 3x = 4C*e 4x - 4C*e 4x + e 3x (3 - 4) = - e 3x Which is equal to the left side of the given equation and therefore y = C*e 4x + e 3x is a solution to the differential equation y ' - 4y = -e 3x. Most of the work on differential equations consists in solving these equations. For example to solve the following differential example y ' = 2x Let us integrate both sides of the given equation as follows ò y ' dx = ò 2x dx which gives y + C1 = x 2 + C2 where C1 and C2 are constants of integration. The solution y of the above equation is given by: y = x 2 + C, where C = C2 - C1.