# What are the applications of differential equations

## Differential equations and their applications

### Chapter 1. First order differential equations

This book deals with differential equations and their applications. A differential equation establishes a relationship between a function and its derivatives. The equations

$$ \ frac {dy} {dt} = 3 {{y} ^ {2}} \ sin \ left (t + y \ right) $$

and $$ \ frac {{{d} ^ {3}} y} {d {{t} ^ {3}}} = {{e} ^ {- y}} + t + \ frac {{{d} ^ { 2}} y} {d {{t} ^ {2}}} $$

are examples of differential equations. The order of a differential equation is the order of the highest derivative of the function y that occurs in the equation. So (i) is a first-order differential equation, (ii) a third-order differential equation. A solution of a differential equation is a continuous function y (t) which, together with its derivatives, satisfies the given relationship. For example, the function solves $$ y \ left (t \ right) = 2 \ sin t - \ frac {1} {3} \ cos 2t $$

the differential equation of the 2nd order $$ \ frac {{{d} ^ {2}} y} {d {{t} ^ {2}}} + y = \ cos2t $$

, because simple recalculation results $$ \ frac {{{d ^ 2}}} {{d {t ^ 2}}} \ left ({2 \, \ sin \; t - \ frac {1} {3} \ cos 2t} \ right ) + \ left ({2 \, \ sin \; t - \ frac {1} {3} \ cos 2t} \ right) = \ left ({- 2 \, \ sin \; t + \ frac {4} {3} \ cos 2t} \ right) + 2 \, \ sin \; t - \ frac {1} {3} \ cos 2t = \ cos 2t $$

.### Chapter 2. Linear differential equations of the second order

A relationship of form

$$ \ frac {{{d} ^ {2}} y} {d {{t} ^ {2}}} = f \ left (t, y, \ frac {dy} {dt} \ right) $$

is called the second order differential equation.### Chapter 3. Systems of differential equations

In this chapter we want to investigate systems of first-order differential equations. Such a system consists of simultaneous first-order differential equations in several variables and has the form

$$ \ begin {gathered} \ frac {{{\ text {d}} {{\ text {x}} _ {\ text {1}}}}}} {{{\ text {dt}}}} {\ text {=}} {{\ text {f}} _ {\ text {1}}} \ left ({{\ text {t,}} {{\ text {x}} _ {\ text {1}} } {\ text {, \ ldots,}} {{\ text {x}} _ {\ text {n}}}} \ right) {\ text {,}} \ hfill \ \ frac {{{\ text {d}} {{\ text {x}} _ {\ text {2}}}}} {{{\ text {dt}}}} {\ text {=}} {{\ text {f}} _ {\ text {2}}} \ left ({{\ text {t,}} {{\ text {x}} _ {\ text {1}}} {\ text {\ ldots,}} {{\ text {x}} _ {\ text {n}}}} \ right) {\ text {, \ ldots,}} \ frac {{{\ text {d}} {{\ text {x}} _ {\ text {n}}}}} {{{\ text {dt}}}} {\ text {=}} {{\ text {f}} _ {\ text {n}}} \ left ({{\ text { t,}} {{\ text {x}} _ {\ text {1}}} {\ text {, \ ldots,}} {{\ text {x}} _ {\ text {n}}}} \ right) \ hfill \ \ end {gathered} $$

### Chapter 4. Qualitative theory of differential equations

In this chapter we consider the differential equation

$$ \ overset {\ centerdot} {\ mathop {x}} \, = f (t, x), $$

in the \ (x = \ left (\ begin {matrix} {{x} _ {1}} (t) \ \ vdots \ {{x} _ {n}} (t) \ end {matrix } \ right) \) and \ (f (t, x) = \ left (\ begin {matrix} {{f} _ {1}} (t, {{x} _ {1}}, \ cdots, {{x} _ {n}}) \ \ vdots \ {{f} _ {n}} (t, {{x} _ {1}}, \ cdots, {{x} _ {n}} ) \ \ end {matrix} \ right) \) a nonlinear function of x_{1}, ..., x_{n}designated. Unfortunately, no methods of solving equation (1) are known. This is of course very regrettable, but in most applications it is not necessary to find the solutions of (1) explicitly. For example, let x_{1}(t) and x_{2}(t) the populations of two species at time t, which are fighting against each other for the food and habitat that is only available to a limited extent in their microcosm. Then the growth rates of x_{1}(t) and x_{2}(t) is described by the differential equation (1), in this case we are less attached to the values of x_{1}(t) and x_{2}(t) at any point in time t, rather than the qualitative properties of x_{1}(t) and x_{2}(t) interested.### Chapter 5. Separation of Variables and Fourier Series

In the applications that we want to examine more closely in this chapter, we are faced with the following problem:

Problem: For which values of λ can we find nontrivial functions y (x) that

$$ \ frac {{{d ^ 2} y}} {{d {x ^ 2}}} + \ lambda y = o; ay \ left (o \ right) + by '(o) = o, cy \ left (\ ell \ right) + dy '\ left (\ ell \ right) = o $$

fulfill? Equation (1) is called the boundary value problem because, in contrast to the initial value problem, in which we have the value of y and y 'at a point x = x_{O}specify the values of the solution y (x) and its derivation y '(x) in two different points x = o and x = ℓ.### Backmatter

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