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Initial Value Problems Differential Equations

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Department
Mathematics
Course
MAT136H1
Professor
Anthony Lam
Semester
Summer

Description
Key Concept Summary Differential Equations are a vast and complex field of study. For the purposes of this course, you will have to deal with first order differential equations. These problems are fairly easy, especially if you have seen them before. Let us do a few practice problems as that may be the best way to understand how to do these problems: Note that a first order differential equation is directly integrable if and only if it can be written as ddy= f(x) Example: Solve the following differential equation 2x + x = x dx 3 dy To solve this differential equation, let us “isolate for  dx: dy 3 2x dxx = x dy 3 2x dx = x − x dy x −x dx= 2x Then we can integrate both sides: dy = x −xdx 2x    x −x ∫dy = ∫ 2x dx       y = ∫ − )dx1   2 2 x3 1 y = 6 − 2 + C Example: If  F(x) is the antiderivative of f(x) = 8x − 1 such that F(1) = 4, then F(2) =:    ⇒ F(x) = 8x∫− 1   2 F(x) = 4x − x + C , but since we know that F(1) = 4, let us substitute that into our equation; 2 4 = 4(1) − (1) + C 1 = C Thus, F(x) = 4x − x + 3 F(2) = 4(2) − 2 + 1 = 15 Related Past Exam Questions: 3 2005, 1) If F(x)is the antiderivative off(x) =  8x + 4x such that F(2) = 43, then F(1)= a) ­12
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