Functions

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1

What Are Functions in Math?

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Why Are Functions Important in Real Life?

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What Is the Definition of a Function?

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What Is a Coordinate System?

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What Are Function Tables?

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How to Find x and y Values of a Function

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How to Calculate Domain and Range in Math

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How to Calculate and Interpret Intersection Points

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How to Calculate Zeros or Roots of a Function

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What Are Linear Functions?

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How Are Linear Functions Used in Real Life?

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What Makes a Function Proportional?

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What Are Quadratic Functions?

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What Are Polynomial Functions?

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What Are Exponential Functions?

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Exponential Functions with Euler's Number

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What Does Inversely Proportional Mean in Math?

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What Are Power and Root Functions?

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What Are Logarithmic Functions?

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What Are Rational Functions?

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Calculate and Interpret Average Rate of Change in Math

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What Is the Point-Slope Equation?

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Interpret and Find Instantaneous Rate of Change in Math

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How to Approximate Instantaneous Rate of Change in Math

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How to Interpret and Calculate Limits of a Function

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How Do You Determine Continuity of a Function?

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How to Interpret and Calculate Asymptotes of a Function

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What Does the Definition of the Derivative Mean?

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How Do You Determine if a Function Is Differentiable?

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Rules of Differentiation for Finding Derivatives

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How Does the Chain Rule for Differentiation Work?

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How to Find Derivatives using the Product Rule

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How to Differentiate Functions Using the Quotient Rule

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Make Sign Charts of the Derivatives of a Function

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Derivation of Distance to Get Velocity and Acceleration

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What Are Stationary Points of a Function?

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How to Calculate the Curvature of a Function

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What Are Inflection Points of a Function?

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How to Analyze Polynomial Functions

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How to Analyze Logarithmic Functions

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How to Analyze Power Functions

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How to Analyze the Cosine Function

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Analyzing Exponential Functions

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What Are Riemann Sums Used for?

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What Are the Important Integration Rules?

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Which Basic Relations in Integration Should You Know?

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How to Interpret and Calculate the Definite Integral

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How to Interpret and Calculate the Indefinite Integral

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How to Use the Formula for Integration by Parts

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How to Do Integration by Substitution

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Use Partial Fraction Decomposition for Integration

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How to Find the Area Between Two Graphs by Integration

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How to Use Solids of Revolution to Find Volumes

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What Is a Differential Equation?

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What Are Integral Curves and Direction Fields Used for?

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What Can Differential Equations Be Used for?

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What Are First Order Differential Equations?

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First Order Differential Equations with Constant Coefficients

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First Order Differential Equations with Initial Conditions

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How to Solve Exact First Order Differential Equations

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How to Write and Solve Separable Differential Equations

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Integrating Factor Method for Differential Equations

The order of a differential equation is determined by the highest order derivative involved! Therefore, first order differential equations consist only of a function and its derivative.

A first order differential equation can be written as

y^{'} + f (x) y = g (x)

You solve these types of equations using integrating factors.

Theory

Integrating Factor

The integrating factor is $e^{F (x)}$ , where

F (x) = \int f (x) d x

and $F^{'} (x) = f (x)$ without $C$ .

By using the product rule, you see that

{(e^{F (x)} y)}^{'} = e^{F (x)} y^{'} + f (x) e^{F (x)} y

This is what you want to introduce into the original equation. Then the differential equation can be solved by integrating factors.

Rule

Instructions for Using the Integrating Factor to Solve Differential Equations

\begin{array}{l} y^{'} + f (x) y & = g (x) | \cdot e^{F (x)} \\ e^{F (x)} y^{'} + f (x) e^{F (x)} y & = {(e^{F (x)} y)}^{'} = e^{F (x)} g (x) \\ \int {(e^{F (x)} y)}^{'} d x & = \int e^{F (x)} g (x) d x \\ e^{F (x)} y & = \int e^{F (x)} g (x) d x \\ y & = \int e^{F (x)} g (x) d x \\ \cdot e^{- F (x)} \end{array}

\begin{array}{l} y^{'} + f (x) y & = g (x) | \cdot e^{F (x)} \\ e^{F (x)} y^{'} + f (x) e^{F (x)} y & = {(e^{F (x)} y)}^{'} = e^{F (x)} g (x) \\ \int {(e^{F (x)} y)}^{'} d x & = \int e^{F (x)} g (x) d x \\ e^{F (x)} y & = \int e^{F (x)} g (x) d x \\ y & = \int e^{F (x)} g (x) d x \cdot e^{- F (x)} \end{array}

Example 1

Solve the differential equation $y^{'} + 2 x y = e^{x}$ by using integrating factors

\begin{array}{l} y^{'} + 2 x y & = e^{x} & | \cdot e^{x^{2}} \\ e^{x^{2}} y^{'} + 2 x e^{x^{2}} y & = 2 x e^{x^{2}} \\ {(e^{x^{2}} y)}^{'} & = 2 x e^{x^{2}} \\ \int {(e^{x^{2}} y)}^{'} d x & = \int 2 x e^{x^{2}} d x \\ e^{x^{2}} y & = \int 2 x e^{x^{2}} d x \\ e^{x^{2}} y & \overset{*}{=} \int 2x e^{u} \frac{1}{2x} d u \\ e^{x^{2}} y & = \int e^{u} d u \\ e^{x^{2}} y & = e^{u} + C \\ e^{x^{2}} y & = e^{x^{2}} + C & | \cdot e^{- x^{2}} \\ y & = 1 + C e^{- x^{2}} \end{array}

\begin{array}{l} y^{'} + 2 x y & = e^{x} & | \cdot e^{x^{2}} \\ e^{x^{2}} y^{'} + 2 x e^{x^{2}} y & = 2 x e^{x^{2}} \\ {(e^{x^{2}} y)}^{'} & = 2 x e^{x^{2}} \\ \int {(e^{x^{2}} y)}^{'} d x & = \int 2 x e^{x^{2}} d x \\ e^{x^{2}} y & = \int 2 x e^{x^{2}} d x \\ e^{x^{2}} y & \overset{*}{=} \int 2x e^{u} \frac{1}{2x} d u \\ e^{x^{2}} y & = \int e^{u} d u \\ e^{x^{2}} y & = e^{u} + C \\ e^{x^{2}} y & = e^{x^{2}} + C & | \cdot e^{- x^{2}} \\ y & = 1 + C e^{- x^{2}} \end{array}

*

\begin{aligned} u & = x^{2} \\ \frac{d u}{d x} & = 2 x \\ d u & = 2 x d x \\ \frac{1}{2 x} d u & = d x \end{aligned}

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63

Logistic Growth and Carrying Capacity of a Population

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What Are Second Order Differential Equations?

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How to Solve Exact Second Order Differential Equations

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Homogeneous Second Order Differential Equations with Constant Coefficients

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Second Order Differential Equations with Initial Conditions

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Verify Solutions to Second Order Differential Equations

69

Find the Equation of a Tangent Line of a Function

70

The Differential Equation for Harmonic Oscillators

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Reduction of Order Method for Differential Equations

72

What Does Mathematical Modeling Mean?

73

What Is Mathematical Regression?

74

The Difference Between Interpolation and Extrapolation

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What Are Linear Models?

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What Are Quadratic Models?

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What Are Exponential Models?

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What Are Power Models in Math?

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What Are Logistic Models Used For?

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What Are Cubic Models in Math?

81

What Are Trigonometric Models?

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What Does the Feasible Region in Optimization Mean?

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How to Use the Graphical Method for Linear Optimization

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How to Use the Ruler Method for Linear Optimization