Ordinary Differential Equations

Q1 - ODE's

Introduction The relationship between independent variable 'x' and dependent variable 'y'. The derivative, dy/dx is the rate of change of something (first order differential equation) The second derivative, d2y/dx2 is the rate of change of the rate of change (second order differential equation) dy/dx is also written as y' d2y/dx2 is also written as y"

Direct Integration When the differential equation can be written as dy/dx = f(x) it can be solved directly by integrating both sides with respect to x This approach can also be used to solve equations of the form d2y/dx2 = f(x) If an additional condition is known, then the integration constant can be determined This means the particular equation can be obtained

Variable Separable Method When the differential equation must be written as dy/dx = f(x)g(y) [where f(x) is a function of x and g(y) is a function of y] it must be solved by separation of variables, first rearranging so that all the y terms are on the left hand side and all the x terms are on the right hand side The general solution (equation containing constants such as 'c') can then be obtained by integrating both sides with respect to x If an additional condition (eg y(2) = 0) is given, it can be used to determine the constant 'c' to obtain the particular solution To answer a question, follow these steps: Separate the variables (get all the x and all the y on separate sides) Integrate, then rearrange to obtain the general solution 'y = ...' Use the given condition to find the particular solution

Integration Factor When the differential equation can be written as dy/dx + p(x)y = q(x) where p(x) and q(x) are different functions of x, it can be solved using the integration factor Both sides of the equation are multiplied by a specially chosen function which allows the left hand sixe to be integrated analytically The integration factor α(x) is such that dα(x)/dx = α(x)p(x) α(x) = e ^ ∫ p(x) dx To answer a question follow these steps: Find the integration factor Multiply both sides of the differential equation by the integration factor Use the product rule to re-write the left hand side Integrate both sides with respect to x, then rearrange to obtain the general solution Use the given condition to obtain the particular solution

Second order linear differential equations Many problems in engineering result in second order differential equations in the form:a d2y/dx2 + bdy/dx + cy =f(x)Where a, b and c are all constant coefficients and f(x) is a function in terms of the variable x onlyThere are two types of second order differential equations: Homogeneous equations Inhomogeneous equations Homogeneous equations When f(x) =0, the equation is known as a homogeneous equation if y = u(x) and y=v(x) are solutions of the equation then y = u + v is also a solution To find the solution to an ODE in the form shown above, the auxiliary equation method is used This is when the equation takes the form where "am2 + bm + c" forms a part. This is called the auxiliary equation and solutions can be found using the quadratic equation When solving a second order ODE using the auxiliary equation, there are three cases to consider: Distinct real roots (b2 > 4ac) Complex roots (b2 Repeated (real) roots (b2 = 4ac) To answer a question, use the following method: Identify the auxiliary equation Obtain the values of m determine if the solution is real, complex or distinct and construct the general solution Use the given conditions to obtain the particular solution

Inhomogeneous Equations When f(x) does not = 0, the equation is an inhomogeneous second order ODE The general solution to an inhomogeneous equation consists of two parts - the complimentary solution (solution to the homogeneous equation) and the particular integral. y = yc + yi Choosing the particular integral: If f(x) is exponential, choose yi to be exponential with the same growth rate as f(x) If f(x) is polynomial, choose yi to be polynomial to the same degree If f(x) contains sine/cosine choose yi to contain sine/cosine with the same period To answer a question use the following method: find the complimentary solution (yc) Select an appropriate particular integral (yi) substitute the particular integral into the equation combine the complimentary function and particular integral to obtain the general solution Use the given conditions to obtain the particular solution