# differential equations

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differential equations

## 4. ODE

### 4.1. first order linear differential equations

4.1.1. standard linear form y' + p(x)y = q(x)

4.1.2. determine the integrating factor e^(int[p(x)]dx)

4.1.2.1. integrating factor

4.1.3. multiply BOTH sides by integrating factor

4.1.4. integrate

### 4.2. second order non-homogeneous constant coefficient differential equations

4.2.1. Vocab

4.2.2. roots of the characteristic equation

4.2.2.1. Real roots

4.2.2.2. Repeated roots

4.2.2.3. Complex roots

4.2.3. for non-homogenous... let the RHS have a coefficient and call it Y. To find this coefficient substitute Y into the originial differential (You will need to find Y' and Y''.)

4.2.3.1. sines and cosines

4.2.3.2. exponential

4.2.3.3. polynomial

4.2.3.4. Superposition principle IF Y has sum of these functions, then solve separately and sum them up

4.2.3.4.1. Product principle Write down all the general entities. Then multiply them to find all terms. Then add a coefficient for each.

4.2.4. we can use the polynomial y and its derivative y' to find the constant coefficients (A and B) if we know initial conditions.

## 5. systems of diffrential equations

### 5.2. inhomogenous systems

5.2.1. ...?

5.2.1.1. same method (of undetermined coefficients)?