One of the most complex and incomprehensible topics of university mathematics is integration and differential calculus. You need to know and understand these concepts, as well as be able to apply them. Many university technical disciplines are tied to differentials and integrals.
Summary of Equations
These equations are one of the most important mathematical concepts in the educational system. A differential equation is an equation that relates independent variables, a function to be found, and derivatives of this function to variables that are considered independent. The differential calculus for finding the function of one variable is called ordinary. If the desired function depends on several variables, then one speaks of a partial differential equation.
In fact, finding a certain answer to the equation reduces to integration, and the method of solution is determined by the type of equation.
1st order equations
A differential equation of the first order is an equation that is able to describe a variable, a desired function, and its first derivative. Such equations can be given in three forms: explicit, implicit, differential.
Concepts needed to solve
The initial condition is to set the value of the desired function for a given value of a variable that is independent.
The solution of a differential equation - any differentiable function, exactly substituted in the original equation, turns it into identically equal. The solution obtained, which is not explicit, is the integral of the equation.
The general solution of differential equations is a function y = y (x; C), which can satisfy the following propositions:
- A function can have only one arbitrary constant C.
- The resulting function should be a solution to the equation for any arbitrary values of an arbitrary constant.
- For a given initial condition, an arbitrary constant can be uniquely determined so that the particular solution obtained will be consistent with a given initial initial condition.
In practice, the Cauchy problem is often used - finding such a solution that is particular and can be compared with the condition set at the beginning.
The Cauchy theorem is a theorem that emphasizes the existence and uniqueness of a particular solution in differential calculi.
Geometric meaning:
- The general solution y = y (x; C) of the equation is the total number of integral curves.
- Differential calculus allows you to connect the coordinates of a point on the XOY plane and the tangent, which is drawn to the integral curve.
- Setting an initial condition means setting a point on a plane.
- To solve the Cauchy problem means that from the whole set of integral curves representing the same solution to the equation, it is necessary to select the only one passing through the only possible point.
- Fulfillment of the conditions of the Cauchy theorem at a point means that an integral curve necessarily passes through the selected point in the plane (moreover, only one).
Separated Variable Equation
By definition, a differential equation is an equation where its right-hand side describes itself or is reflected in the form of a product (sometimes a relationship) of two functions, one that depends only on "x" and the other only on "y". A clear example for this kind is: y '= f1 (x) * f2 (y).
To solve equations of a specific form, it is first necessary to transform the derivative y '= dy / dx. Then, using manipulations with the equation, it is necessary to bring it to such a form when two parts of the equation can be integrated. After the necessary transformations, we integrate both parts and simplify the result.
Homogeneous Equations
By definition, a differential equation can be called homogeneous if it has the following form: y '= g (y / x).
In this case, the substitution y / x = t (x) is most often used.
To solve such equations, it is necessary to reduce the homogeneous equation to a form with separable variables. To do this, perform the following operations:
- Display, expressing the derivative of the original function, from any source in the form of a new equation.
- The next step is to convert the resulting function to the form f (x; y) = g (y / x). In simpler words, make sure that the equation contains only the ratio y / x and constants.
- Make the following change: y / x = t (x); y = t (x) * x; y '= t' * x + t. The substitution made will help to divide the variables in the equation, gradually leading to a simpler form.
Linear equations
The definition of such equations is as follows: a linear differential equation is an equation where its right-hand side is expressed as a linear expression relative to the original function. The desired function in this case: y '= a (x) * y + b (x).
We rephrase the definition as follows: any first-order equation will become linear in form if the original function and its derivative from it are included in the equation of the first degrees and are not multiplied by each other. The "classical form" of a linear differential equation has the following structure: y '+ P (x) y = Q (x).
Before solving such an equation, it should be converted to a "classical form." The next step will be the choice of a solution method: the Bernoulli method or the Lagrange method.
Solving the equation using the method introduced by Bernoulli involves substituting and reducing the linear differential equation to two equations with separate variables for the comparative functions U (x), as well as V (x), which were given in their original form.
The Lagrange method is to find a general solution to the original equation.
- You should find the same solution to the homogeneous equation. After the search, we have the function y = y (x, C), where C is an arbitrary constant.
- We are searching for a solution to the original equation in the same form, but consider C = C (x). We substitute the function y = y (x, C (x)) into the original equation, find the function C (x) and write the solution to the general initial equation.
Bernoulli equation
Bernoulli equation - if the right-hand side of the calculus takes the form f (x; y) = a (x) y + b (x) yk, where k is any possible rational numerical value, without taking as an example the cases when k = 0 and k = 1.
If k = 1, then the calculus takes the form with separable variables, and for k = 0 the equation remains linear.
Consider the general case of solving this type of equation. We have the standard Bernoulli equation. It needs to be reduced to linear, for this you need to divide the equation by yk. After this operation, replace z (x) = y1-k. After a series of transformations, the equation will be reduced to linear, most often by the substitution method z = U * V.
Equations in Complete Differentials
Definition An equation having the structure P (x; y) dx + Q (x; y) dy = 0 is called an equation in full differentials if the following condition is met (in this condition “d” is a partial differential): dP (x ; y) / dy = dQ (x; y) / dx.
All the previously considered first-order differential equations can be displayed in the form of differentials.
Such calculi are solved in several ways. But, however, they all begin with checking that the condition is fulfilled. If the condition is satisfied, then the leftmost region of the equation is the total differential of the unknown function U (x; y). Then, in accordance with the equation, dU (x; y) will be zero, and therefore the same integral of the equation in the full differentials will be displayed in the form U (x; y) = C. Therefore, solving the equation reduces to finding the function U (x; y )
Integrating factor
If the condition dP (x; y) / dy = dQ (x; y) / dx is not satisfied in the equation, then the equation does not have the form that we considered in the paragraph above. But sometimes it is possible to choose some function M (x; y), when multiplied by which the equation takes the form of an equation in complete "diffusions". The function M (x; y) is referred to as an integrating factor.
Integrating can be found only in those cases when it becomes a function exclusively for one variable.