Page 98 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Unit 5: Differential Equations

Differentiating again we have: Notes
2
d y x 3x
Ae 9Be ...(iv)
dx 2
Subtracting (iii) from (iv) we have

2
d y dy 3x
6Be ...(v)
dx 2 dx
Subtracting (ii) from (iii)

dy 3x
y 2Be ...(vi)
dx
2
d y dy dy
Thus 2 3 3y
dx dx dx
2
d y dy
or 2 4 3y 0 ...(vii)
dx dx
3. The equation of an ellipse is

x 2 y 2
1 ...(viii)
a 2 b 2
Find its differential equation
Differentiating (viii) we get

2x 2y dy 0
a 2 b 2 dx ...(ix)
Differentiating again, we have

2
2 2 dy 2 2y d y
2
a 2 b 2 dx b dx 2 0 ...(x)
From (ix), we have

b 2 y dy
a 2 x dx ...(xi)

From (x) we have

2
b 2 dy 2 d y
y ...(xii)
a 2 dx dx 2
From (xi) and (xii) we have
2 2
dy d y dy y
y
dx dx 2 dx x

2
d y dy 2 y dy
or y 2 0 ...(xiii)
dx dx x dx

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