Unit10 - Unit 10 First Order Dierential Equations with...

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Unit 10 First Order Differential Equations with Applications First Order Differential Equations Definition 10.1. A differential equation is an equation which relates some unknown function, say y , to one or more of its derivatives. Definition 10.2. A first order differential equation involves only the first derivative of the function. A higher order differential equation involves higher derivatives. For instance, a differential equation relating y to its second derivative (and perhaps its first derivative as well) would be a second order differential equation. Examples: y dy dx = x and y - 3 dy dx + 4 = 0 are first order differential equations; also, y + dy dx - d 2 y dx 2 = 3 x 2 - 2 is a second order differential equation. We will sometimes use the abbreviation D.E. for the phrase differential equa- tion. Definition 10.3. A solution to a D.E. is a function of the form y = f ( x ) (i.e., an expression for y which does not involve y or its derivatives), which satisfies the differential equation. We often denote such a solution as y ( x ). Generally, there are many solutions to a differential equation. For instance, the second order D.E. y = - d 2 y dx 2 is satisfied by y = sin x or by y = cos x , or in fact by y = A sin x + B cos x for any values of A and B. To “solve” a D.E. 1
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usually means to find the General Solution, that is, to find the form that all solutions to the D.E. have. We will only be concerned with first order differential equations. We will study two types of first order differential equations, namely separable , and linear differential equations. We need different approaches for solving these 2 different types of D.E.’s. Definition 10.4. A separable first order differential equation is an equation which may be put into one of the following forms: dy dx = f ( x ) g ( y ) or dy dx = g ( y ) f ( x ) or dy dx = f ( x ) g ( y ) Strictly speaking, we have already dealt with some separable D.E.’s, as the first example will show. First, though, we outline some basic steps for solving such a problem. Steps For Solving a Separable Differential Equation: (Assuming that the D.E. is given in terms of dy dx ) 1. Recognize the problem as a separable D.E. e.g. we have dy dx = f ( x ) g ( y ) 2. Separate the equation into the form (terms involving y )( dy ) = (terms not involving y )( dx ) e.g. rearrange dy dx = f ( x ) g ( y ) to g ( y ) dy = f ( x ) dx 3. Integrate both sides, adding one arbitrary constant, say C, to the x side. (This is done because adding an arbitrary constant to both sides after integrating is equivalent to adding just a single arbitrary constant to one side.) e.g. evaluate integraltext g ( y ) dy = integraltext f ( x ) dx 4. Solve for y if possible. If we can solve for y we get what is called the general solution , and if we cannot solve for y we get what is called an implicit (general) solution . 2
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Example 1 . Find the general solution to dy dx = x 2 .
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