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Unformatted text preview: MATH 150 Separable Equations Cheung Man Wai Tutorial 2 An equation is said to be separable if it can be written in the form M ( x ) dx + N ( y ) dy = 0 . Moreover, if we require the initial condition, namely y ( x ) = y , then we can solve it as Z x x M ( x ) dx + Z y y N ( y ) dy = 0 . The crucial step of this method is to rewrite the equation as the form (1) and perform integration thereafter. Now we turn to some examples. Example 1: dy dt = ry , where r is a constant. This eqation is usually called the exponential equation. dy dt = ry Z dy y = Z rdt ln | y | = rt + C y = De rt , where C,D are constants. Example 2: Solve y = 1 + y 2 with y (0) = 1. By separating valuables, we have Z y 1 dy 1 + y 2 = Z x dx tan- 1 y | y 1 = x | x tan- 1 y- π 4 = x y = tan( x + π 4 ) . 1 Note that tan is not defined at π 2 + 2 nπ , for n is any integer. And as we have initial value for x = 0, we want- π 2 < x + π 4 < π 2 ....
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