# Differential Equation/Examples

## Examples of Differential Equations

### Second Order Linear Ordinary Differential Equation

Ordinary linear differential equation of the $2$nd order:

- $\dfrac {\d^2 y} {\d x^2} + y = x^2$

### First Order First Degree Non-Linear Ordinary Differential Equation $(1)$

Ordinary non-linear differential equation of the $1$st order and $1$st degree:

- $\paren {x + y}^2 \dfrac {\d y} {\d x} = 1$

### First Order First Degree Non-Linear Ordinary Differential Equation $(2)$

Ordinary non-linear differential equation of the $1$st order and $1$st degree:

- $\dfrac {\d y} {\d x} = \dfrac x {y^{1/2} \paren {1 + x^{1/2} } }$

### First Order First Degree Non-Linear Ordinary Differential Equation $(3)$

Ordinary non-linear differential equation of the $1$st order and $1$st degree:

- $\dfrac {\d y} {\d x} = 1 + x y^2$

### First Order Second Degree Non-Linear Ordinary Differential Equation

Differential Equation/Examples/First Order Second Degree Non-Linear Ordinary

### Second Order Non-Linear Ordinary Differential Equation

Ordinary non-linear differential equation of the $2$nd order:

- $\dfrac {\d^2 y} {\d x^2} + \paren {3 \dfrac {\d y} {\d x} }^3 + 2 x = 7$

### Second Order Second Degree Non-Linear Ordinary Differential Equation

Ordinary non-linear differential equation of the $2$nd order and $2$nd degree:

- $\paren {1 + \paren {\dfrac {\d y} {\d x} }^2}^{3/2} = 3 \dfrac {\d^2 y} {\d x^2}$

### Third Order Linear Ordinary Differential Equation

Ordinary linear differential equation of the $3$rd order:

- $2 \dfrac {\d^3 y} {\d x^3} + 3 \dfrac {\d^2 y} {\d x^2} + \dfrac {\d y} {\d x} - 10 y = e^{-3 x} \sin 5 x$

### Third Order Second Degree Non-Linear Ordinary Differential Equation

Ordinary non-linear differential equation of the $3$rd order and $2$nd degree:

- $\paren {\dfrac {\d^3 y} {\d x^3} }^2 + \paren {\dfrac {\d^2 y} {\d x^2} }^4 + \dfrac {\d y} {\d x} = x$

### Fourth Order First Degree Non-Linear Ordinary Differential Equation

Ordinary non-linear differential equation of the $4$th order and $1$st degree:

- $x \dfrac {\d^4 y} {\d x^4} + 2 \dfrac {\d^2 y} {\d x^2} + \paren {x \dfrac {\d y} {\d x} }^5 = x^3$

### First Order Linear Partial Differential Equation

Partial linear differential equation of the $1$st order in $2$ independent variables:

- $x \dfrac {\partial z} {\partial x} + y \dfrac {\partial z} {\partial y} - z = 0$

### Second Order Linear Partial Differential Equation $(1)$

Partial linear differential equation of the $2$nd order in $3$ independent variables:

- $\dfrac {\partial^2 V} {\partial x^2} + \dfrac {\partial^2 V} {\partial y^2} + \dfrac {\partial^2 V} {\partial z^2} = 0$

### Second Order Linear Partial Differential Equation $(2)$

Partial linear differential equation of the $2$nd order in $2$ independent variables:

- $\dfrac {\partial^2 y} {\partial t^2} = \alpha^2 \dfrac {\partial^2 y} {\partial x^2}$

### Second Order Second Degree Non-Linear Partial Differential Equation

Partial non-linear differential equation of the $2$nd order and $2$nd degree in $2$ independent variables:

- $\dfrac {\partial^2 z} {\partial x^2} \cdot \dfrac {\partial^2 z} {\partial y^2} - \paren {\dfrac {\partial^2 x} {\partial x \partial y} }^2 = 0$

### First Order First Degree Total Differential Equation

Total differential equation of the $1$st order and $1$st degree:

- $u \rd x + v \rd y + w \rd z = 0$

### First Order Second Degree Total Differential Equation

Total differential equation of the $1$st order and $2$nd degree:

- $x^2 \rd x^2 + 2 x y \rd x \rd y + y^2 \rd y^2 - z^2 \rd z^2 = 0$