“Are you afraid of differential equations?”

This page was thought for a purpose, so for students and people who like to learn about differential equations as definitions, examples, and others.

In a first-order linear differential equation is obtained by replacing the coefficients in the equation by arbitrary functions of “t”;

you must convert the equation into one intgrable by using the product rule for derivatives were the μ ( t ) is called integrating factor and determine how to find it for a given equation.

Example:

The first-order differential equation is:

To identify this class of equation, we first rewrite the first order equation in the form

Where M is a function of x only and N is a function of y only, then the equation becomes

Such an equation is said to be separable, because if it is written in the differential form.

A separable equation can be solved by integration the functions M and N.

Example:

An important class of first-order equations consists of those in which the independent variable does not appear explicitly. Such equations are called autonomous and have the form:

Population Dynamics: exponential growth

Let y=0(t), be the population of a given species at time t;

where

r: rate of growth/decay

r>0, exponential growth model

r<0, exponential decay

Population Dynamics: logistic growth

where a real number of c is a critical point of an autonomous equation.

Where;

Example:

The Euler’s method is a first-order numerical procedure for solving ordinary differential equations (ODE) with a given initial value.

To construct the tangent at the point x and obtain the value of y(x+h), whose slope is,

In Euler’s method, you can approximate the curve of the solution by the tangent in each interval (that is, by a sequence of short line segments), at steps of h.

In general, if you use small step size, the accuracy of approximation increases.

Example:

The Existence and Uniqueness Theorem tells us that the integral curves of any differential equation satisfying the appropriate hypothesis, cannot cross. If the curves did cross, we could take the point of intersection as the initial value for the differential equation.

The initial value problem is:

Demonstrating the Picard-Lindelöf existence and uniqueness theorem isn't an easy task; it will first require developing a preliminary theory in which we establish some new concepts and also briefly review concepts that we already know and will be useful for proving this theorem. This preliminary theory will be developed throughout this and the following entry, culminating in the proof of the theorem in the final entry of this first unit.

We'll begin by stating the Picard-Lindelöf existence and uniqueness theorem to keep it in mind. Even though some things might not be clear initially, the aim of this preliminary theory is to understand what this theorem is telling us, as well as to provide us with the necessary tools to prove it.

Video examples:

Many second-order differential equations have the form:

where f is the given function,

That is, if f is linear in y and dy/dt. In equation, and q are specified functions of the independent variable t but do not depend on y. In this case we usually rewrite the equation as:

Where the primes denote differentiation with respect to t . Instead of the previous equation, we sometimes see the equation

Now we can divide the equation by P(t) and thereby obtain this equation with:

An initial value problem consists of a differential equation such as equations (1), (3), or (4) together with a pair of initial conditions

A second-order linear differential equation is said to be homogeneous

The functions P, Q, and R are constants. In this case, the equation becomes:

Example:

If Y1 and Y2 are two solutions of the non-homogeneous linear differential equation, then their difference Y1 – Y2 is a solution of the corresponding homogeneous differential equation, as in the second one. If, in addition, Y1 and Y2 form a fundamental set of solutions of the second equation, then.

Where c1 and c2 are certain constants. The general solution of the non-homogeneous equation can be written as:

Where Y1 and Y2 form a fundamental set of solutions of the corresponding homogeneous equation, c1 and c2 are arbitrary constants, and Y is any solution of the first non-homogeneous equation.

Example:

Is the general method for finding a particular solution of a differential equation by replacing the constants in the solution of a related homogeneous equation by functions and determining these functions so that the original differential equation will be satisfied.

To illustrate the method, suppose it is desired to find a particular solution of the equation

To use this method, it is necessary first to know the general solution of the corresponding homogeneous equation i.e., the related equation in which the right-hand side is zero. If y1(x) and y2(x) are two distinct solutions of the equation, then any combination;

The variation of parameters consists of replacing the constants an and b by functions u1(x) and u2(x) and determining what these functions must be to satisfy the original non-homogeneous equation. After some manipulations, it can be shown that if the functions u1(x) and u2(x) satisfy the equations:

will satisfy the original differential equation. These last two equations can be solved to give:

These last equations either will determine u1 and u2 or else will serve as a starting point for finding an approximate solution.

Video example:

An nth order linear differential equation of the form:

Theorem 1: For nth order differential equation with initial conditions:

Linearly dependence and Independence. The functions are said to be linearly dependent on an interval I if there exists a set of constants not all zero.

is called a fundamental set of solutions if

If is a fundamental set, then:

its called the general solution of the differential equation.

The method of variation of parameters for determining a particular solution of the non-homogeneous nth order linear differential equation.

Is a direct extension of the method for the second-order differential equation, the method of variation of parameters is still more general than the method of undetermined coefficients in that it leads to an expression for the particular solution for any continuous function g, whereas the method of undetermined coefficients is restricted in practice to a limited class of functions g.

Suppose then that we know a fundamental set of solutions y1, y2, . . . , yn of the homogeneous equation. Then the general solution of the homogeneous equation is:

The method of variation of parameters for determining a particular solution of the initial equation rests on the possibility of determining n functions u1,u2, ... ,un such that Y(t) is of the form:

Since we have n functions to determine, we will have to specify n conditions. One of these is clearly that Y satisfy the given equation. The other n − 1 conditions are chosen so as to make the calculations as simple as possible. Since we can hardly expect a simplification in determining Y if we must solve high order differential equations for u1, . . . , un, it is natural to impose conditions to suppress the terms that lead to higher derivatives of u1, . . . , un. From last equation we obtain:

where we have omitted the independent variable t on which each function in the last equation depends. Thus the first condition that we impose is that,

It follows that the expression of the equation for Y ′ reduces to:

We continue this process by calculating the successive derivatives Y′′, ... ,Y(n−1). After each differentiation we set equal to zero the sum of terms involving derivatives of u1, . . . , un. In this way we obtain n − 2 further conditions similar to the last equation; that is,

As a result of these conditions, it follows that the expressions for Y′′, . . . , Y(n−1) reduce to

Finally, we need to impose the condition that Y must be a solution of the given equation. By

differentiating Y(n−1) from the last equation, we obtain:

To satisfy the differential equation we substitute for Y

By solving this system and then integrating the resulting expressions, you can obtain the coefficients u1, . . . , un. Using Cramer’s3 rule, we can write the solution of the system of equations in the form

With this notation a particular solution of the given equation is given by

Where t0 is arbitrary.

An infinite series of the form

where 𝑥0 and 𝑎0, 𝑎1, …, 𝑎𝑛, …are constants, is called a power series in 𝑥−𝑥0. We say that the power series the previous equation converges for a given 𝑥 if the limit

exists; otherwise, we say that the power series diverges for the given 𝑥.

The next theorem shows that if the power series converges for some 𝑥≠𝑥0 then the set of all values of 𝑥 for which it converges forms an interval.

For any power series

exactly one of the these statements is true:

I. The power series converges only for 𝑥=𝑥0.

II. The power series converges for all values of 𝑥.

III. There’s a positive number 𝑅 such that the power series converges if |𝑥−𝑥0|<𝑅 and diverges if |𝑥−𝑥0|>𝑅.

The next theorem provides a useful method for determining the radius of convergence of a power series. It’s derived in calculus by applying the ratio test to the corresponding series of absolute values.

Suppose there’s an integer 𝑁 such that 𝑎𝑛≠0 if 𝑛≥𝑁 and

where 0≤𝐿≤∞. Then the radius of convergence of ∑∞𝑛=0𝑎𝑛(𝑥−𝑥0)𝑛 is 𝑅=1/𝐿, which should be interpreted to mean that 𝑅=0 if 𝐿=∞, or 𝑅=∞ if 𝐿=0.

To practice more on the topic, click here

In this section we will begin to consider how to solve equations of the form

Euler Equations: A relatively simple differential equation that has a singular point is the Euler equation

Equal Roots: A second solution can be obtained by the method of reduction of order, but for the purpose of our future discussion we consider an alternative method.

Solutions of an Euler equation; real roots (μ = 0).

The two typical second solutions of an Euler equation with equal roots: r > 0(red),r < 0(blue).

The general solution of the Euler equation:

in any interval not containing the origin is determined by the roots r1 and r2 of the equation

as follows. If the roots r1 and r2 are real and different, then,

If the roots are real and equal, then

If the roots are complex conjugates, then,

The solutions of an Euler equation of the form