Recitation 14

First order linear equation

Constant coefficients The general solution to y'=ay+b, where a and b are constants, is y=\frac{Ae^{ax}-b}{a}. Here A is a constant that can be determined given an initial condition.

Integrating factor The big idea of integrating factor is to multiply both sides of y'+p(x)y=q(x) by \mu(x) = \exp \int p(x)dx so that the left hand side becomes (\mu y)'.

Separable equation A separable equation is an equation of the form a(x)dx = b(y)dy. Let A(x), B(y) be the anti-derivative of a(x), b(y). Then A(x) = B(y) + C for some constant C depending on the initial condition.

Exact equation Suppose we have the differential equation M(x,y)+N(x,y)y'=0 and a region D. If there is a function \Psi(x,y) so that \Psi_x(x,y) = M(x,y), \Psi_y = N(x,y) for all (x,y)\in D, then we call the differential equation exact in D. In this case, the implicit solution is \Psi(x,y) = c for (x,y)\in D, and we call \Psi(x,y) the potential function.

Test for exact differential equation Suppose the region D is simply connected. The differential equation M(x,y)+N(x,y)y'=0 is exact in D if and only if M_y(x,y) = N_x(x,y) at each (x,y)\in D.

Integrating factor + exact equation It is sometimes possible to convert a differential equation that is not exact into an exact equation by multiplying the equation by a suitable integrating factor.

Existence and uniqueness Suppose that F(x,y) is a continuous function defined in some region R = (x_0-\delta, x_0+\delta)\times (y_0-\epsilon, y_0+\epsilon) containing the point (x_0, y_0). Then there exists \delta_1 > 0 so that a solution y=f(x) to y'=F(x,y) is defined for x\in (x_0-\delta_1, x_0+\delta_1). Suppose, furthermore, that \frac{\partial F}{\partial y}(x,y) is a continuous function defined on R. Then there exists \delta_2 > 0 so that the solution is the unique solution to y'=F(x,y) for x\in (x_0-\delta_2, x_0 + \delta_2).

Second order linear equation

Homogeneous with constant coefficients The characteristic equation of ay'' + by' + c=0 is ar^2 + br + c = 0. Let r_1, r_2 be the roots of the characteristic equation. (1) When r_1, r_2 are distinct reals, the general solution is y = c_1e^{r_1x} + c_2e^{r_2x}; (2) When r_1, r_2 = \lambda \pm i\mu are complex, the general solution is y=e^{\lambda x}(c_1\cos \mu x + c_2\sin \mu x); (3) When r_1, r_2 = r are repeated roots, the general solution is y=e^{rx}(c_1 + c_2x.

Non-homongenous The general solution of the second order nonhomogeneous linear equation y'' + p(x)y' + q(x)y = g(x) can be expressed in the form y = y_h + y_p, where y_p is any particular function that satisfies the nonhomogeneous equation and y_h is a general solution to the homogeneous equation y'' + p(x)y' + q(x)y = 0.

Higher order linear equation

Homogeneous with constant coefficients The characteristic equation of y^{(n)} + a_1y^{(n-1)} + a_2y^{(n-2)} + \dots + a_ny = 0 is r^n + a_1r^{n-1} + a_2r^{n-2} + \dots + a_n = 0. If the roots of the characteristic equation are distinct reals r_1, r_2, \dots, r_n, then the general solution is c_1e^{r_1x} + c_2e^{r_2x} + \dots + c_ne^{r_nx}.

Existence and uniqueness Given a linear differential equation y^{(n)} + p_1(x)y^{(n-1)} + p_2(x)y^{(n-2)} + \dots + p_n(x)y = g(x) with initial conditions y(x_0)=y_0, y'(x_0)=y_1, \dots, y^{(n-1)}(x_0)=y_{n-1}, x_0 \in (x_1, x_2). If p_1, \dots, p_n, g are continuous on the open interval (x_1, x_2), then there exists exactly one solution to the initial value problem for x_1 < x < x_2.

Wronskian For n functions f_1(x), \dots, f_n(x), the Wronskian W(f_1, \dots, f_n) is defined by \begin{vmatrix} f_{1}(x) & f_{2}(x) & \cdots & f_{n}(x) \\ f_{1}'(x) & f_{2}'(x) & \cdots & f_{n}'(x)\\ \vdots & \vdots & \ddots & \vdots \\ f_{1}^{(n-1)}(x) & f_{2}^{(n-1)}(x) & \cdots & f_{n}^{(n-1)}(x) \end{vmatrix}. (1) If the Wronskian of n functions is not identically zero, then these n functions are linearly independent. (2) If the Wronskian of n solutions to an nth order homogeneous linear differential equation does not vanish on an interval (x_1, x_2), then these solutions form a fundamental set of solutions.

Order reduction Order reduction is employed when one solution y_1(x) is known and other linearly independent solutions are desired. Assume the other solutions are of the form y = vy_1. Plugging this substitution into the differential equation then leads to a linear differential for v of a lower order.

Radius of convergence of series solutions If the Taylor series of p_1(x), p_2(x), \dots, p_n(x) converge for |x-x_0| < r, then the series solutions to y^{(n)} + p_1(x)y^{(n-1)} + p_2(x)y^{(n-2)} + \dots + p_n(x)y = 0 converge for |x-x_0|<r. In other words, the radius of convergence of the series solutions is at least the minimum radius of convergence of p_1(x), p_2(x), \dots, p_n(x).