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Unformatted text preview: Notes for Day : § . : Examp es of Di erentia Equations Introduction A di erential equation is an equation that may involve not only an independent variable x and a dependent variable y , but also one or more derivatives of y with respect to x . For example, x + sin y- y ′ = is a di erential equation. From this point on however, we will generally use t , and not x , to denote the independent variable. When we solve a di erential equation, we are nding a function y = f ( t ) which makes the equation true. Examp e: Show that the function y = sin t is a solution to the di erential equation t y + y ′′ = cos t Examp e : Gravity Here’s a di erential equation you’ve seen before: Consider a falling object. We know that its acceleration due to gravity is- . m/s , and we also know that acceleration is the second derivative of position. e di erential equation, then, is: y ′′ ( t ) = - ....
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