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Question

# Explain the basic concepts ordinary and partial differential equations (ODEs, PDEs), order, general and particular solutions, initial value problems (IVPs). Give examples.

Solution

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If the derivatives in the equation are ordinary, the equation is said to be ordinary (ODE).

Example:

$y' + y = 0$

$\frac{dy}{dx} + (x+ 15) y = 0$

If the derivatives in the equation have refrence tot two or more independent variables, the equation is said to be partial (PDE).

Example:

$x \frac{d^2y}{dx^2} + (2x-1)y \frac{dy}{dx} + xy = 0$

If the $n$th derivative is the highest derivative in the equation, that equation is said to be of order '$n$'.

Example:

$y''' + 3y'' - 3y' + y = 30e^x$

Order of this differential equation is $3$.

$y^{\text{iv}} + 10 y '' - 9y = 0$

Order of this differential equation is $4$.

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