# First and Second Order Differential Equations ## First Order Differential equations

A first order differential equation is of the form: ### Linear Equations: The general general solution is given by where is called the integrating factor.

### Separable Equations: (1)
Solve the equation g(y) = 0 which gives the constant solutions.
(2)
The non-constant solutions are given by ### Bernoulli Equations: (1)
Consider the new function .
(2)
The new equation satisfied by v is (3)
Solve the new linear equation to find v.
(4)
Back to the old function y through the substitution .
(5)
If n > 1, add the solution y=0 to the ones you got in (4).

### Homogenous Equations: is homogeneous if the function f(x,y) is homogeneous, that is By substitution, we consider the new function The new differential equation satisfied by z is which is a separable equation. The solutions are the constant ones f(1,z) - z =0 and the non-constant ones given by Do not forget to go back to the old function y = xz.

### Exact Equations: is exact if The condition of exactness insures the existence of a function F(x,y) such that All the solutions are given by the implicit equation ## Second Order Differential equations

### Homogeneous Linear Equations with constant coefficients: Write down the characteristic equation (1)
If and are distinct real numbers (this happens if ), then the general solution is (2)
If (which happens if ), then the general solution is (3)
If and are complex numbers (which happens if ), then the general solution is where that is ### Non Homogeneous Linear Equations: The general solution is given by where is a particular solution and is the general solution of the associated homogeneous equation In order to find two major techniques were developed.

### Method of undetermined coefficients or Guessing Method

This method works for the equation where a, b, and c are constant and where is a polynomial function with degree n. In this case, we have where The constants and have to be determined. The power s is equal to 0 if is not a root of the characteristic equation. If is a simple root, then s=1 and s=2 if it is a double root.
Remark. If the nonhomogeneous term g(x) satisfies the following where are of the forms cited above, then we split the original equation into N equations then find a particular solution . A particular solution to the original equation is given by ### Method of Variation of Parameters

This method works as long as we know two linearly independent solutions of the homogeneous equation Note that this method works regardless if the coefficients are constant or not. a particular solution as where and are functions. From this, the method got its name.
The functions and are solutions to the system: which implies Therefore, we have ### Euler-Cauchy Equations: where b and c are constant numbers. By substitution, set then the new equation satisfied by y(t) is which is a second order differential equation with constant coefficients.

(1)
Write down the characteristic equation (2)
If the roots and are distinct real numbers, then the general solution is given by (2)
If the roots and are equal ( ), then the general solution is (3)
If the roots and are complex numbers, then the general solution is where and . [Differential Equations]
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