Linear Equations

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Modeling with Differential Equations

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Euler's Method and Direction Fields

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Euler's method revisited

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The Simplest Differential Equations
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Linear Equations

Linear ODEs: Working an Example
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Calculus with Parametrized Curves

Video: Slope and Area
Video: Arclength and Surface Area
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Higher Derivatives

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Video: Computing Slopes of Tangent Lines

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Convergence of Series with Negative Terms

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Strategy to Test Series and a Review of Tests
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Power Series

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Representing Functions as Power Series

Functions as Power Series
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Taylor Polynomials
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Functions of 2 and 3 variables

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Partial Derivatives

One variable at a time (yet again)
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An Example from DNA
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Differentiability and the Chain Rule

Differentiability
The First Case of the Chain Rule
Chain Rule, General Case
Video: Worked problems

Multiple Integrals

General Setup and Review of 1D Integrals
What is a Double Integral?
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Iterated Integrals over Rectangles

How To Compute Iterated Integrals
Examples of Iterated Integrals
Fubini's Theorem
Summary and an Important Example

Double Integrals over General Regions

Type I and Type II regions
Examples 1-4
Examples 5-7
Swapping the Order of Integration
Area and Volume Revisited

Double integrals in polar coordinates

dA = r dr (d theta)
Examples

Multiple integrals in physics

Double integrals in physics
Triple integrals in physics

Integrals in Probability and Statistics

Single integrals in probability
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Change of Variables

Review: Change of variables in 1 dimension
Mappings in 2 dimensions
Jacobians
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Bonus: Cylindrical and spherical coordinates

In General

In the following video, we work out the solution to a general linear first order ODE $$\frac{dy}{dx} + P(x) y = Q(x),$$ and apply the method to the example $$\frac{dy}{dx} + 2xy = 4x.$$

We always take our integrating factor to be $$I(x) = e^{\int P(x) dx}.$$ (The method doesn't depend on which constant of integration we choose. Just pick any anti-derivative of $P(x)$.)

Notice that $$I'(x) = P(x) I(x),$$ so $$\left(I(x)y\right)' = I(x) y' + I'(x) y = I(x)\left(y' + P(x) y\right).$$
Multiplying our ODE by $I(x)$ gives $$\left(I(x) y\right)' = I(x) Q(x).$$

Integrate both sides to get $$I(x) y = \int I(x) Q(x) \,dx.$$ This involves a constant of integration, which we'll write explicitly as $+C$.

Divide by $I(x)$ to get our general solution: $$y = \frac{1}{I(x)} \left ( \int I(x) Q(x)\, dx + C\right ).$$

Plug in the initial conditions to figure out what $C$ is.

In our example, $y' + 2xy = 4x$, since $P(x) = 2x$ we can take $I(x) = e^{x^2}$. Hence $$y = \frac{1}{e^{x^2}} \int 4x e^{x^2} dx +C = \frac{1}{e^{x^2}} (2e^{x^2}+C) = 2 + C e^{-x^2}.$$