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Lecture-6

# Lecture-6 - AMS 361 Applied Calculus IV(DE BVP Outline for...

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AMS 361: Applied Calculus IV (DE & BVP) Outline for Lecture 6 6. Substitution Methods 6.1 Basic Concepts and for simplest case: DE y’= f(x,y) can be solved by Substitution Methods. e.g. 1. Solve DE y’ = (x+y+3)^2. We can easily define a new variable v = x+y+3, which leads to y = v-x-3, thus, dy/dx = dv/dx -1. Thus, the equation becomes dv/dx -1 = v^2 or Becomes dv/dx = v^2 +1. Therefore, Thus, v = tan (x +C) x+y+3 = tan (x +C) Therefore, y = tan(x+C) – x – 3. In general, all equations y’=F(ax+by+c) can be solved this way. SHOW Details to class. First, you introduce a new variable v = ax+by+c. Next, take derivative wrt to x on both sides: or So the original equation can be transformed to or Therefore, we can find the equation as or

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6.2 Homogeneous Equations y’ = F(y/x) Homogeneous Equations can be solved by substitution v = y/x. Then, y =v x è Therefore, the original equation becomes or or This equation can be solved easily by separation of variable! e.g. 2: Solving DE: 2xy y’ = 4x^2 + 3y^2 or This is a simple Homogeneous equation. So, we just follow the above steps to solve this quickly.
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Lecture-6 - AMS 361 Applied Calculus IV(DE BVP Outline for...

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