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Unformatted text preview: MATH 150 Introduction to Ordinary Differential Equations Cheung Man Wai Tutorial 1 1 Introduction An ordinary differential equation (or ODE) is a relation that contains functions of only one independent variable, and one or more of its derivatives with respect to that variable. ODE arises in many different areas other than mathematics, e.g. physics, modelling. 2 Differentiation Suppose f is a differentiable function. The derivative of y = f ( x ), denoted as f ( x ) or dy dx , is defined as the slope of the tangent line to the curve y = f ( x ) at the point ( x, y ). Mathematically, it is defined as the limit f ( x ) = dy dx = df ( x ) dx = lim h → f ( x + h )- f ( x ) h . 2.1 Differentiation formulae Instead of getting into limiting everytime, we have several rules to evaluate the derivatives of combinations of simple functions. For the proof of these rules, please refer http://www.math.ust.hk/ machas/calculus.pdf. Suppose f , g are differentiable functions. Then (a) ( f + cg ) ( x ) = f ( x ) + cg ( x ), where c is a constant....
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