Introduction to Differential Equations
What is a differential equation?
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:
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 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.
Differential Equations
