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Classification of Differential Equations

Partial differential equations (PDE):
A differential equation that contains partial derivatives is called a partial differential equation. Note that a PDE has more than one independent variables. For example, \begin{align*} \frac{\partial u}{\partial t}=k\frac{\partial^2u}{\partial x^2}\text{ (heat equation)} && (5) \end{align*}

Ordinary differential equations (ODE):
A differential equation that does not contain partial derivatives is called an ordinary differential equation. Note that an ODE has only one independent variable. For example, \begin{align*} \frac{dv}{dt}=g-\frac{\gamma}{m}v \text{ (motion of falling objects)} && (6) \end{align*} The order of a differential equation is the the order of the highest derivative in it. Differential equation (6) has order 1. Differential equation (5) has order 2.

For an ODE, if the independent variable is $$t$$, then it is sometimes written as $F(t,y(t),y',y'',\ldots,y^{(n)})=0$ $\text{or, }y^{(n)}=f(t,y(t),y',y'',\ldots,y^{(n-1)}).$

Linear ODE: An ODE $$F(t,y(t),y',y'',\ldots,y^{(n)})=0$$ is linear if $$F$$ is a linear function of $$t,y(t),y',y'',\ldots,y^{(n)}$$. It can be written as $a_0(t)y^{(n)}+a_1(t)y^{(n-1)}+\cdots+a_n(t)y=g(t).$ For example, $ty''+y\sin t=e^t$ is a linear ODE of order 2. But the following 2nd order ODEs are nonlinear. \begin{align*} ty''+yy'&=e^t\\ ty''+y^2&=e^t\\ ty''+\sin(t+y)&=e^t \end{align*}

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