Feel free to use and distribute these tutorials, keeping each intact. Click on any of the tutorials below to launch it. In mathematics, linear differential equations are differential equations having solutions which can be added second order homogeneous differential equation pdf in particular linear combinations to form further solutions. Such an equation is said to have order n, the index of the highest derivative of y that is involved.

A typical simple example is the linear differential equation used to model radioactive decay. 0 is called a homogeneous equation and its solutions are called complementary functions. When the Ai are numbers, the equation is said to have constant coefficients.

The first method of solving linear homogeneous ordinary differential equations with constant coefficients is due to Euler, who realized that solutions have the form ezx, for possibly-complex values of z. The exponential function is one of the few functions to keep its shape after differentiation, allowing the sum of its multiple derivatives to cancel out to zero, as required by the equation. 0 is the characteristic equation considered later by Gaspard Monge and Augustin-Louis Cauchy.

Solving the polynomial gives n values of z, z1, , zn. Substitution of any of those values for z into ezx gives a solution ezix. Since homogeneous linear differential equations obey the superposition principle, any linear combination of these functions also satisfies the differential equation.