Types of Solutions of a Differential Equations

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Types of Solutions of a Differential equation

A differential equation is a mathematical equation for an unknown function of one or several

variables that relates the values of the function itself and its derivatives of various orders.Differential equations play a prominent role in engineering, physics, economics, and other disciplines.

Differential equations are mathematically studied from several different perspectives, mostlyconcerned with their solutions²the set of functions that satisfy the equation. Only the simplestdifferential equations admit solutions given by explicit formulas; however, some properties of solutions of a given differential equation may be determined without finding their exact form. If a self-contained formula for the solution is not available, the solution may be numericallyapproximated using computers. The theory of dynamical systems puts emphasis on qualitativeanalysis of systems described by differential equations, while many numerical methods have

been developed to determine solutions with a given degree of accuracy.

A relation y = (x) is said to be a solution or integral of the differential equation f (x, y, y', . . . , y{sup (n)}) = 0 in the range a x b if, when substituted into the equation, it gives a result thatis identically zero in that range (see 169). It is frequently difficult or undesirable to express asolution y explicitly as a function of x, but instead to have an implicit relation F (x, y) = 0between the solution y and the independent variable x. Such a relation is a solution if, whensolved explicitly for y in terms of x, it yields a solution in the way described above. For example,if both sides of a polynomial equation in x and y are differentiated with respect to x, then thefunction y(x) determined by this implicit relation will be a solution of the differential equation

thereby obtained (see 170). The simplest of all ordinary differential equations is y' = g (x), inwhich g is an elementary function. An example is g (x) = 2x. The problem of solving thisdifferential equation is equivalent to finding a function y(x) the derivative of which is 2x; thisleads to the solution y = x + c in which c is an arbitrary constant. The process of integration hasled to the occurrence of an arbitrary constant in the solution. It can be seen intuitively (and it canbe proved rigorously) that finding the solution of a differential equation of the nth order willsomehow be equivalent to performing n integrations and that the final solution y(x) will contain n arbitrary constants of integration.

Any solution of an nth order equation of the form y = (x, c , c , . . . , c ) in which c , . . . , c are arbitrary constants is called a general solution of the equation. Any solution that may beobtained from the general solution of an equation by assigning particular values to the constantsis called a particular solution of that equation. It sometimes happens that a nonlinear differentialequation has a solution that cannot be obtained by assigning specific values to the arbitraryconstants in the general solution. Such a solution is called a singular solution of the differentialequation.

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For example, a particular nonlinear differential equation (see 171) has a general solution that islinear in the independent variable (see 172) with an arbitrary constant c. A particular solutionmay be obtained from (172) by taking c = 1 (see 173). On the other hand, a function proportionalto x (see 174) also satisfies the differential equation. Because this solution cannot be obtainedfrom the general solution by assigning a particular value to c, it is a singular solution of the

differential equation (171). In this instance the graphs of the functions defined by (172) arestraight lines, and the graph of the function defined by (174) is a parabola that is the envelope of that family of straight lines (cf. Figure 18).

Similarly, a nonlinear equation with the cube of the derivative proportional to y (see 175) has ageneral solution with y proportional to (x - c) (see 176) and the singular solution y = 0. Herethe general solution is represented by a family of semicubical parabolas, and the singular solution is the line that passes through the cusp of each member of the family (cf. Figure 19).Such a line is called a cusp locus.

The differential equations are very much helpful in many areas of science. But most of

interesting real life problems involve more than one unknown function. Therefore, the use of system of differential equations is very useful. Without loss of generality, we will concentrate onsystems of two differential equations

As a motivation let us consider an island with two type of species: Rabbits and Fox. Clearly oneplays the role of predator while the other one the role of a prey. If we are interested to model thepopulations growths of both species, then we have to keep in mind that if, for example, thepopulation of the Fox increases, then the Rabbit population will be affected. So the rate of change of the population of one type will depend on the actual population of the other type. For example, in the absence of the Rabbit population, the Fox population will decrease (and fast) toface a certain extinction. Something that most of us would like to avoid. A model for thisPredator-Prey problem was developed by Lotka (in 1925) and Volterra (in 1926) and is known asthe Lotka-Volterra system

where R(t ) measures the Rabbit population, F (t ) measures the Fox population, and all the

involved constant are positive numbers. Note that a and b are the growth rate of the



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prey, and the death rate of the predator. and are measures of the effect of the interactionbetween the Rabbits and The Fox.

Note that in the Lotka-Volterra system, the variable t is missing. This kind of system is calledautonomous system and are written

Phase Plane

Let us go back to the general case

A solution to this system is the couple of functions (x(t ),y(t )) which satisfy both differentialequations of the system. When we change the variable t , then we get a set of points on the xy-plane which, in physics, we usually call a trajectory. The moving object has the coordinates

(x(t ),y(t )) at time t . The velocity to the trajectory at time t is given by

Note that we do not need to know the solution (x(t ),y(t ) to determine the velocity vector at time t .Indeed, we have

as long as we know x, y, and t . In particular, we can draw all the velocity vectors everywhere onthe plane for the autonomous systems. This is known as the vector field.The vector field should be understood as the analogue of the direction field for differentialequations.

Example. Draw the vector field for the predator-prey problem



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A differential equation (or "DE") contains derivatives or differentials.

Recall from the Differential section in the Integration chapter, that a differential can be thoughtof as a derivative where "d y/ d x" is not written in fraction form.

Ex

amples of Differentials

d x (this means "an infinitely small change in x")

d (this means "an infinitely small change in ")

d t (this means "an infinitely small change in t ")

Examples of Differential Equations

Example 1.

We saw the following example in the Introduction to this chapter. It involves a derivative(d y/ d x):

As we did before, we would integrate it to produce the required solution. This will be a general

solution (involving K , a constant of integration).

So we proceed as follows:

and this gives



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But where did that d y go from the d y/ d x? Why did it seem to disappear?

In this example, we appear to be integrating the x part only (on the right), but in fact we haveintegrated with respect to y as well (on the left). Differential equations are like that - you need tointegrate with respect to two (sometimes more) different variables, one at a time.

We could have written our question only using differentials:

d y = (x2 í 3)d x

(All I did was to multiply both sides of the original d y/ d x in the question by d x.)

Now we integrate both sides, the left side with respect to y (that's why we use "d y") and the right

side with respect to x (that's why we use "d x") :

� d y = �(x2 í 3)d x

Then the answer is the same as before, but this time we have arrived at it considering the d y partmore carefully:

On the left hand side, we have integrated � d y = � 1 d y to give us y.

Note about the constant: We have integrated both sides, but there's a constant of integration onthe right side only. What happened to the one on the left? The answer is quite straightforward.We do actually get a constant on both sides, but we can combine them into one constant ( K ) which wewrite on the right hand side.

Example 2

This example also involves differentials:

2d = sin(t + 0.2) d t

We have:

A function of with d on the left side, and

A function of t with d t on the right side.



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To solve this, we would integrate both sides, one at a time, as follows:

� 2d = � sin(t + 0.2) d t

3/3 = ícos(t + 0.2) + K

We have integrated with respect to on the left and with respect to t on the right.

Solving a differential equation

From the above examples, we can see that solving a differential equation means finding an equationwith no derivatives that satisfies the given differential equation. Solving a differential equation alwaysinvolves one or more integration steps.

It is important to be able to identify the type of differential equation we are dealing with before weattempt to solve it.

Definitions

First order DE: Contains only first derivatives

Second order DE: Contains second derivatives (and possibly first derivatives also)

Degree: The highest power of the highest derivative which occurs in the differential equation.

Examples - Order and Degree

1)

This DE has order 2 (the highest derivative appearing is the second derivative) and degree 1 (the power of the highest derivative is 1.)

2)

This DE has order 1 (the highest derivative appearing is the first derivative) and degree 5 (the power of the highest derivative is 5.)



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3) ( y " ) 4 + 2( y ' ) 7 í 5y = 3

This DE has order 2 (the highest derivative appearing is the second derivative) and degree 4 (the power of the highest derivative is 4.)

General and Particular Solutions

When we first performed integrations, we obtained a general solution (involving a constant, K ).

We obtained a particular solution by substituting known values for x and y. These known conditions arecalled boundary conditions (or initial conditions).

It is the same concept when solving differential equations - find general solution first, then substitute

given numbers to find particular solutions.

Let's see some examples of finding solutions of first order, first degree DEs.

Example 1

a. Find the general solution for the differential equation

d y + 7x d x = 0.

b. Find the particular solution given that y(0) = 3.

Answer

a. We simply need to subtract

7x d x

from both sides, then insert integral signs and integrate:

NOTE 1: We could have written it in a more familiar way as:



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Then So

NOTE 2: means . We also have:

and so on.

b. We now use the information y(0) = 3 to find K .

The information means that at x = 0, y = 3. We substitute these values into the equation that we found inpart (a), to find the particular solution.

gives K = 3.

So the particular solution is:

Here is the graph of our solution.

Example 2

Find the particular solution of

y' = 5



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given that when x = 0, y = 2.

Answer

We can write

y' = 5

as a differential equation:

d y = 5 d x

Integrating both sides gives:

y = 5x + K

Applying the boundary conditions: x = 0, y = 2, we have K = 2 so:

y = 5x + 2

Example 3

Find the particular solution of

y''' = 0

given that:

y(0) = 3, y' (1) = 4, y'' (2) = 6

Answer

Since y''' = 0, when we integrate once we get:

y'' = A (A is a constant)

Integrating again gives:

y' = Ax + B (A, B are constants)

Once more:

y = Ax2/2 + Bx + C (A, B and C are constants)

The boundary conditions are:



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y(0) = 3, y' (1) = 4, y'' (2) = 6

We need to substitute these values into our expressions for y'' and y' and our general solution, y = Ax2/2 +Bx + C.

Now

y(0) = 3 gives C = 3

and

y'' (2) = 6 gives A = 6 (actually, y'' = 6 for any value of x in this problem since there is no x term)

Finally,

y' (1) = 4 gives B = -2

So the particular solution for this question is:

y = 3x2 í 2x + 3

Checking the solution by differentiating and substituting initial conditions:

y' = 6x í 2

y' (1) = 6(1) í 2 = 4

y'' = 6

y''' = 0

Our solution is correct.

Example 4

After solving the differential equation,

(we will see how to solve this DE in the next section Separation of Variables), we obtain the result

y = c ln x

Did we get the correct general solution?



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Answer

Now, if y = c ln x, then

[See Derivative of the Logarithmic Function if you are rusty on this.)

So

We conclude that we have the correct solution.

Second Order DEs

We include two more examples here to give you an idea of second order DEs. We will see later in thischapter how to solve such Second Order Linear DEs.

Example 5

The general solution of the second order DE

y'' + a2y = 0

is

y = A cos ax + B sin ax



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Example 6

The general solution of the second order DE

y'' í 3y' + 2y = 0

is

y = Ae2x + Be

x

If we have the following boundary conditions:

y(0) = 4, y' (0) = 5

then the particular solution is given by:

y = e

2x

+ 3e

x

Now we do some examples using second order DEs. We are given a solution and we need tocheck if it is the correct solution.

Example 7 - Second Order DE

Show that

y = c1 sin 2x + 3 cos 2x is a general solution for the differential equation

Answer

We have a second order differential equation and we have been given the general solution. Our

job is to show that the solution is correct.

We do this by substituting the answer into the original 2nd order differential equation.

We need to find the second derivative of y:

y = c1 sin 2x + 3 cos 2x



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First derivative:

Second derivative:

Now for the check step:

Example 8 - Secon

dOr

der D

E

Show that has a solution of y = c1 + c2e2x

Answer

Since

y = c1 + c2e2x, then:



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and

It is obvious that

Differential equations arise in many areas of science and technology, specifically whenever adeterministic relation involving some continuously varying quantities (modeled by functions)and their rates of change in space and/or time (expressed as derivatives) is known or postulated.

This is illustrated in classical mechanics, where the motion of a body is described by its positionand velocity as the time varies. Newton's laws allow one to relate the position, velocity,acceleration and various forces acting on the body and state this relation as a differentialequation for the unknown position of the body as a function of time. In some cases, thisdifferential equation (called an equation of motion) may be solved explicitly.

An example of modelling a real world problem using differential equations is determination of the velocity of a ball falling through the air, considering only gravity and air resistance. Theball's acceleration towards the ground is the acceleration due to gravity minus the decelerationdue to air resistance. Gravity is constant but air resistance may be modelled as proportional to theball's velocity. This means the ball's acceleration, which is the derivative of its velocity, depends

on the velocity. Finding the velocity as a function of time involves solving a differentialequation.

Types of Solutions of a Differential Equations

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