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StudyGuide250Exam1 - Separable Equations dy/dx =...

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Separable Equations: dy/dx = f(y)g(x) (take integral, and separate as follows): F(y) + c1 = G(x) + c2 Also, you want to determine the intervals of validity (where the solution is not undefined) and the final interval of validity is the one containing the initial value of x. Linear First Order Equations: dy/dt + p(t)y(t) = q(t) It is absolutely essential that the equation be in this form before we try to solve it. The method is completely dependant on the coefficient of y0 being 1 and p(t) having the correct sign. Notice that if p(t) = 0 or q(t) = 0, the equation is not only linear, but separable as well. However, if p(t) doesn’t = 0 and q(t) doesn’t = 0, separation of variables will not work. First order differential equations relate the slope of a function to the values of the function and the independent variable Integrating Factors: (1) Put the equation in the correct form y’ + p(t)y = q(t) (2) Calculate the integrating factor (µ(t) = e^Integral( p(t) dt) (3) Multiply both sides of the equation by µ(t). (4) Integrate both sides, being careful not to forget the constant of integration. (5) Solve for y(t), using the initial condition (if applicable) to calculate the constant of integration.
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