Differential equations
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Transcript of Differential equations
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Differential Equations
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Separable Differential Equations
A separable differential equation can be expressed as the product of a
function of x and a function of y.
dyg x h y
dx
Example:
22dy
xydx
Multiply both sides by dx and divide both sides by y2 to separate the variables. (Assume y2 is never zero.)
22
dyx dx
y
2 2 y dy x dx
0h y
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A separable differential equation can be expressed as the product of a
function of x and a function of y.
dyg x h y
dx
Example:
22dy
xydx
22
dyx dx
y
2 2 y dy x dx
2 2 y dy x dx
21x C
y
2
1y
x C
0h y
Separable Differential Equations
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Example:
222 1 xdyx y e
dx
2
2
12
1xdy x e dx
y
Separable differential equation
2
2
12
1xdy x e dx
y
2u x
2
1
1udy e du
y
Solve the differential with initial condition
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Example (cont.):
222 1 xdyx y e
dx
21tan xy e C We now have y as an implicit function of x.
We can find y as an explicit function of x by taking the tangent of both sides.
21tan tan tan xy e C
2
tan xy e C
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A harder problem
Careful when separating, wecan only separateby multiplyingor dividing!!!
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A Trickier problem!