470.mws Series Solutions near an Ordinary Point Example: Solve y''+y=0. Example 2: Find a series solution in powers of x of Airy's equation y''-xy=0. Example 3: Solve Airy's equation in powers of x-1. Example 4: Solve Airy's equation in powers of x-1. Finding Radius of Convergence of Series Solution Example 1. What is the radius of convergence of the Taylor series for 1/(x^2-2*x+2) about x=0? About x=1? Example 2: Determine a lower bound for the radius of convergence of series solutions about x=0 and x=-1/2 for the different2 for the differential equation (1+x^2)*y^"+2*x*y^`'`+4*x^2*y=0 . Regular Singular Points Example: Determine the singular points of the differential equation(x+2)^2*(x-1)*diff(y(x),x,x)+3*(x-1)*diff(y(x),x)-2*(x+2)*y = 0;. Example: Determine the singular points of (x-Pi/2)^2*diff(y(x),x,x)+cos(x)*diff(y(x),x)+sin(x)*y = 0; . Series Solutions near a Regular Singular Point Theorem: Example: Bessel's Equation Bessel's Equation of Order Zero, v=0 Bessel's Equation of Order Zero, v = 1/2; Bessel's equation of order 1 Laplace Transform Example 1: Solve diff(y(x),x,x)+y(x)=sin(2*t) with initial conditions y(0)=2 and diff(y(0),x)=1 . Example 2. Solve 2*diff(y(x),x,x)+diff(y(x),x)+2*y(x) = g(t); where g(t) = Heaviside(t-5)-Heaviside(t-20); with initial conditions y(0) = 0; and diff(y(0),x) = 0; . Systems of First Order Linear Equations Example 1: Solve the system diff(x(t),t)=MATRIX([[-3, sqrt(2)], [sqrt(2), -2]])*x(t). Example 2: Solve the system diff(x(t),t)=MATRIX([[0, 1, 1], [1, 0, 1], [1, 1, 0]])*x(t) . In the case when eigenvalues are complex we have to find real and imaginary parts of the complex-valued solutions which we obtain in the same way as in the real case. Example 4: Solve the system diff(x(t),t)=MATRIX([[1,- 1], [1, 3]])*x(t) . Example 5: Solve the system diff(x(t),t)=MATRIX([[5,-3,-2], [8,-5,-4], [-4,3,3]])*x(t) .

tml>