Second Order Equations

STEP 1999, Math I, #7: Show that , where is a constant, satisfies the differential equation

(*)

In the particular case when

, find the solution of the equation (*) of the form

and at .

Use this result to express

and satisfy the differential equation

show that and .

Show also that satisfies the differential equation for any constants and .

Given instead that and satisfy the equation (*), find and .

STEP 2007, Math III, #7: (i) Find functions and such that and both satisfy the equation

For these functions and , write down the general solution of the equation.

Show that the substitution

transforms the equation

(*)

into

and hence show that the solution of equation (*) that satisfies at is given by

(ii) Find the solution of the equation

STEP 2009, Math III, #2: (i) Let

where the coefficients

are independent of and are such that this series and all others in this question converge.

Show that

and write down a similar expression for .

Write out explicitly each of the three series as far as the term containing

.

(ii) It is given that

satisfies the differential equation

By substituting the series of part (i) into the differential equation and comparing coefficients, show that

.

Show that, for

,

and , then .

Find the corresponding result when

and .

STEP 2010, Math I, #6: Show that, if , then

In order to find other solutions of this differential equation, now let , where is a function of .

By substituting this into (*), show that

(**)

By setting

in (**) and solving the resulting first order differential equation for

, find in terms of . Hence show that

satisfies (*), where and are any constants.

STEP 2011, Math III, #1: (i) Find the general solution of the differential equation

(ii) Show that substituting (where is a function of ) into the second order differential equation

leads to a first order differential equation for . Find and hence show that the general solution of (*) is

where

and are arbitrary constants.

(iii) Find the general solution of the differential equation

STEP 2012, Math III, #1: Given that

show that

(i) Use the above result to show that the solution to the equation

and when is

.

(ii) Find the solution to the equation

of the drug in the blood and the brain respectively are and . These satisfy

,

where the dot denotes differentiation with respect to t .

Obtain a second order differential equation for

and hence derive the solution

(i) Obtain the solution that satisfies and . The quantity of the drug in the brain for this solution is denoted by .

(ii) Obtain the solution that satisfies , where is a given constant. The quantity of the drug in the brain for this solution is denoted by .

(iii) Show that for

,

provided

takes a particular value that you should find.

STEP 1999, Math III, #8: The function is defined for and satisfies the conditions

is in the range , where is a positive integer, satisfies the differential equation

(ii) Show that for , and find for all .

(iii) Show that

Constant Coefficients

A second order differential equation with constant coefficients is one that takes the form

where

, and are real numbers. We can find a complete set of two independent solutions for this equation by using the technique of guessing the form of the solutions and substituting that form back into the equation.

Let's assume that the solution takes the form

This means that and so substituting these into the differential equation we get

This equation will be satisfied if and only if

this is known as the characteristic equation of the differential equation above. Since it's just a quadratic, we can find its solutions.

There are three cases to consider. The roots of the characteristic equation are either:

1.) Two distinct real numbers

and in which case our general solution becomes

2.) Two non-real, complex conjugates and in which case our solutions become (by way of Euler's equation)

3.) One double root

in which case our solutions become