# Differential Equations

Topics covered in a first year course in differential equations. Need to understand basic differentiation and integration from Calculus before starting here.

• ##### Introduction to Differential Equations
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What a differential equation is and some terminology?

• ##### Separable Differential Equations
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Introduction to separable differential equations.

• ##### Separable Differential Equations II
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Another separable differential equation example.

• ##### Exact Equations Intuition Part 1 (proofy)
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Chain rule using partial derivatives (not a proof; more intuition).

• ##### Exact Equations Intuition Part 2 (proofy)
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More intuitive building blocks for exact equations.

• ##### Exact Equations Example 1
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First example of solving an exact differential equation.

• ##### Exact Equations Example 2
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Some more exact equation examples.

• ##### Exact Equations Example 3
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One more exact equation example.

• ##### Integrating Factors 1
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Using an integrating factor to make a differential equation exact.

• ##### Integrating Factors 2
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Now that we've made the equation exact, let's solve it!

• ##### First Order Homogeneous (Part 1)
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Introduction to first order homogenous equations.

• ##### First Order Homogeneous (Part 2)
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Another example of using substitution to solve a first order homogeneous differential equations.

• ##### 2nd Order Linear Homogeneous Differential Equations 1
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Introduction to 2nd order, linear, homogeneous differential equations with constant coefficients.

• ##### 2nd Order Linear Homogeneous Differential Equations 2
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Let's find the general solution!

• ##### 2nd Order Linear Homogeneous Differential Equations 3
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Let's use some initial conditions to solve for the particular solution.

• ##### 2nd Order Linear Homogeneous Differential Equations 4
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Another example with initial conditions!

• ##### Complex Roots of the Characteristic Equations 1
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What happens when the characteristic equations has complex roots?!

• ##### Complex Roots of the Characteristic Equations 2
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What happens when the characteristic equation has complex roots?

• ##### Complex Roots of the Characteristic Equations 3
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Lets do an example with initial conditions!

• ##### Repeated Roots of the Characteristic Part 1
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What happens when the characteristic equation only has 1 repeated root?

• ##### Repeated Roots of the Characteristic Part 2
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An example where we use initial conditions to solve a repeated-roots differential equation.

• ##### Undetermined Coefficients 1
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Using the method of undetermined coefficients to solve nonhomogeneous linear differential equations.

• ##### Undetermined Coefficients 2
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Another example using undetermined coefficients.

• ##### Undetermined Coefficients 3
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Another example where the nonhomogeneous part is a polynomial.

• ##### Undetermined Coefficients 4
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Putting it all together!

• ##### Laplace Transfrom 1
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Introduction to the Laplace Transform.

• ##### Laplace Transfrom 2
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Laplace transform of e^at.

• ##### Laplace Transfrom 3
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Laplace Transform of sin(at) (part 1).

• ##### Laplace Transfrom 4
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Part 2 of getting the Laplace transform of sin(at).

• ##### Laplace Transfrom 5
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Useful properties of the Laplace Transform.

• ##### Laplace Transform 6
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Laplace transform of cosine and polynomials!

• ##### Laplace Transform to Solve an Equation
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Using the Laplace Transform to solve an equation we already knew how to solve.

• ##### Laplace Transform Solves an Equation 2
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Second part of using the Laplace Transform to solve a differential equation.

• ##### More Laplace Transform Tools
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A grab bag of things to know about the Laplace Transform.

• ##### Using the Laplace Transform to Solve a Nonhomogeneous Equation
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Solving a non-homogeneous differential equation using the Laplace Transform.

• ##### Laplace Transform of: L(t)
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Determining the Laplace Transform of t.

• ##### Laplace Transform of t^n: L(t^n)
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Laplace Transform of t^n: L{t^n}.

• ##### Laplace Transform of the the Unit Step
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Introduction to the unit step function and its Laplace Transform.

• ##### Inverse Laplace Examples
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Using our toolkit to take some inverse Laplace Transforms.

• ##### Laplace/Step Function Differential
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Hairy differential equation involving a step function that we use the Laplace Transform to solve

• ##### Dirac Delta Function
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Introduction to the Dirac Delta Function.

• ##### Laplace Transform of the Dirac Delta
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Figuring out the Laplace Transform of the Dirac Delta Function.

• ##### Introduction to the Convolution
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Introduction to the Convolution.

• ##### The Convolution and the Laplace
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Understanding how the product of the Transforms of two functions relates to their convolution.

• ##### Using the Convolution Theorem to Solve an Intial Value Prob
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Using the Convolution Theorem to solve an initial value problem.