warm-up problems solve the ivp. give the largest interval over which the solution is defined
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Warm-up Problems
Solve the IVP . Give the largest interval over which the solution is defined.
2 1 2 6 , 0 5dydxx xy x y
Exact EquationsRecall that we defined the differential of
z = f (x,y) as dz = fx dx + fy dy
Ex. Consider x2 – 5xy + y3 = c, find the differential of both sides.
M(x,y)dx + N(x,y)dy is an exact differential if it is the differential of some function f (x,y). A DE of the form M(x,y)dx + N(x,y)dy = 0 is called an exact equation.
But how can we identify an exact equation?
Thm. M(x,y)dx + N(x,y)dy is an exact differential iff
N M
x y
Ex. Solve 2xy dx + (x2 – 1)dy = 0.
Ex. Solve
(e2y – y cos xy)dx + (2xe2y – x cos xy + 2y)dy = 0.
Ex. Solve 2
2
cos sin, 0 2
1
dy xy x xy
dx y x
To make M(x,y)dx + N(x,y)dy exact, we multiply by an integrating factor:
If is in terms of only x, use
If is in terms of only y, use
y xM N
N
y xM Ndx
Ne
x yN M
M
x yN Mdy
Me
Ex. Solve xy dx + (2x2 + 3y2 – 20)dy = 0.