Integral Calculus - What Is 1St Order Differential Equation

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Topic: Integral Calculus
Subtopic: What is a 1st order differential equation?

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A differential equation is an equation that relates a function with its derivatives.
A first order differential equation involves only the first derivative of the unknown function.

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A general form of a first order differential equation is: dy/dx = f(x,y)
Here, y is the unknown function and f(x,y) represents the relationship between x, y, and their derivatives.

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Example: dy/dx = x + y
This is a first order differential equation as it involves only the first derivative.
The equation represents a relationship between the derivative of y with respect to x and the variables x and y.

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Solutions to a first order differential equation can be found by integrating both sides of the equation.
Integrating the equation dy/dx = f(x,y) with respect to x gives: ∫ (dy/dx) dx = ∫ f(x,y) dx

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Integrating both sides of the equation dy/dx = x + y with respect to x gives: ∫ dy/dx dx = ∫ (x + y) dx
Simplifying the integration, we get: y = 1/2 x^2 + C
Here, C is the constant of integration.

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The solution y = 1/2 x^2 + C is called the general solution of the first order differential equation dy/dx = x + y.
The constant of integration C represents the family of all possible solutions to the differential equation.

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To find a particular solution to the differential equation, we need to consider an initial condition.
An initial condition specifies the value of the unknown function at a certain point.
For example, if we are given y(0) = 2, we can substitute this initial condition into the general solution to find the particular solution.

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Substituting y(0) = 2 into the general solution y = 1/2 x^2 + C, we get: 2 = 1/2 (0)^2 + C
Simplifying further, we find: C = 2
Therefore, the particular solution to the first order differential equation dy/dx = x + y with the initial condition y(0) = 2 is: y = 1/2 x^2 + 2

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In conclusion, a first order differential equation involves only the first derivative of the unknown function.
Solutions to first order differential equations can be found by integrating both sides of the equation.
The constant of integration represents the family of all possible solutions.
To find a particular solution, an initial condition is required.

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Integrating Factors
An integrating factor is a function that is used to transform a linear first order differential equation into an exact differential equation.
It helps in simplifying the process of finding the solution to the differential equation.
The integrating factor is typically denoted by the symbol µ.
To find the integrating factor, we need to identify the coefficient of the dependent variable in the differential equation.

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Example:
Consider the first order differential equation: dy/dx + 2y = 4x.
Here, the coefficient of y is 2.
To find the integrating factor, we divide the coefficient of y by x.
Integrating factor µ = e^(∫2dx) = e^(2x)
Multiplying the differential equation by the integrating factor, we get: e^(2x) dy/dx + 2e^(2x) y = 4x e^(2x)

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Solving Exact Differential Equations
An exact differential equation is a type of differential equation in which the total differential of the unknown function can be written in the form dF(x,y) = 0.
These equations have a unique solution.
To solve an exact differential equation, we need to find a function F(x,y) such that its total derivative yields the given equation.
This can be done by equating the coefficient of dx in the given equation to the derivative of F(x,y) with respect to x, and the coefficient of dy to the derivative of F(x,y) with respect to y.

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Example:
Consider the differential equation: (2x + y) dx + (x + 3y) dy = 0.
To check if it is exact, we equate the coefficient of dx to the derivative of a function F(x,y) with respect to x, and the coefficient of dy to the derivative of F(x,y) with respect to y.
We have: (∂F/∂x) = 2x + y and (∂F/∂y) = x + 3y.
Integrating (∂F/∂x) with respect to x gives: F = x^2 + xy + g(y), where g(y) is the integration constant.
Differentiating F with respect to y and equating it to (∂F/∂y), we can find g(y).

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Solving Differential Equations by Separation of Variables
The method of separation of variables is used to solve a first order ordinary differential equation.
This method involves isolating the variables x and y on separate sides of the equation and integrating each side separately.
By integrating both sides, we can find a general solution to the differential equation.

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Example:
Consider the differential equation: dy/dx = y/x.
To solve this equation using separation of variables, we can rewrite it as: dy/y = dx/x
Integrating both sides, we get: ln|y| = ln|x| + c, where c is the constant of integration.
Exponentiating both sides, we have: |y| = |x| * e^c
Removing the absolute values, we get the general solution: y = kx, where k = ±e^c.

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Homogeneous Differential Equations
A homogeneous differential equation is a type of differential equation where all terms can be written in the same degree of the dependent variable.
By making a substitution and simplifying, these equations can be transformed into separable form.
Homogeneous differential equations are solved using substitution techniques.

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Example:
Consider the differential equation: dy/dx = (2x^2 + y^2) / (xy).
To solve this equation, we make the substitution: y = ux.
Taking the derivative of y with respect to x, we have: dy/dx = u + x(du/dx)
Substituting the values in the original equation and simplifying, we get a separable equation in terms of u and x.
Solving the separable equation, we find the particular solution.

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Linear Differential Equations
A linear differential equation is a type of differential equation that can be written in the form: dy/dx + P(x)y = Q(x).
These equations can be solved using integrating factors or by the method of variation of parameters.
Linear differential equations have general solutions that involve arbitrary constants.

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Example:
Consider the linear differential equation: dy/dx - 2y = x^2.
To solve this equation using an integrating factor, we identify P(x) = -2 and Q(x) = x^2.
The integrating factor µ = e^(∫-2dx) = e^(-2x).
Multiplying the differential equation by the integrating factor, we get: e^(-2x) dy/dx - 2e^(-2x)y = x^2e^(-2x).
Integrating both sides and solving for y, we find the general solution.

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Second Order Differential Equations
A second order differential equation is an equation that relates a function with its second derivatives.
It involves the unknown function, its first derivative, and its second derivative.
A general form of a second order differential equation is: d^2y/dx^2 = f(x, y, dy/dx)
Here, y is the unknown function and f(x, y, dy/dx) represents the relationship between x, y, and their derivatives.

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Example: d^2y/dx^2 + 2dy/dx + y = 0
This is a second order differential equation as it involves the second derivative, first derivative, and the variable y.
The equation represents a relationship between the second derivative of y with respect to x, the first derivative, and y.

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Solutions to a second order differential equation can be found by techniques such as:
- Method of undetermined coefficients
- Variation of parameters
- Power series method
- Laplace transforms
The choice of method depends on the given equation and initial conditions.

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Method of Undetermined Coefficients
This method is used to find the particular solution of a non-homogeneous second order linear differential equation.
It involves assuming a particular form for the solution based on the non-homogeneous term and finding the unknown coefficients.
The general solution of the equation is the sum of the particular solution and the complementary solution obtained from the associated homogeneous equation.

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Variation of Parameters
Variation of parameters is used to find the particular solution of a non-homogeneous second order linear differential equation.
It involves assuming the particular solution as a linear combination of two linearly independent solutions of the associated homogeneous equation.
By equating coefficients and solving the resulting equations, the particular solution is obtained.

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Power Series Method
The power series method is used to find a solution to a second order linear differential equation in the form of a power series.
The power series is centered around a particular point, and the coefficients of the series are determined by substituting it into the differential equation.
This method is particularly useful for solving differential equations with variable coefficients.

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Laplace Transforms
Laplace transforms can be used to solve initial value problems of second order linear differential equations.
The Laplace transform of a differential equation converts it into an algebraic equation, which can be solved using algebraic methods.
After finding the inverse Laplace transform of the solution, the particular solution is obtained.

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Example: Solve the second order differential equation d^2y/dx^2 + 2dy/dx + y = 2e^x
We can use the method of undetermined coefficients in this case.
For the non-homogeneous term 2e^x, we assume a particular solution of the form yp = Ae^x.
Substituting this into the differential equation and solving for the unknown coefficient A gives the particular solution.

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Example: Solve the second order differential equation d^2y/dx^2 + 2dy/dx + y = 0
This is a homogeneous equation, so finding the solution involves solving the associated characteristic equation.
The characteristic equation is obtained by substituting y = e^(mx) into the differential equation and solving for m.
The solutions of the characteristic equation give the complementary solution of the differential equation.

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In conclusion, second order differential equations involve the second derivative of the unknown function.
There are various methods for finding the solutions to second order differential equations, depending on the form of the equation and initial conditions.
The method of undetermined coefficients, variation of parameters, power series method, and Laplace transforms are some of the techniques used.
The choice of method depends on the given equation and the type of solutions required.