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 = vx3, 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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 Summer '08
 Staff
 Elementary algebra, original equation, simple Homogeneous equation

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