Question

hii

Find the order of given Differential equation :-

y = A sinx + B cos(x+c)

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Answer 1

$\bold{Answer :}$

Given that,

y = A sinx + B cos (x + c) ...(i)

To find the required differential equation and its order, we need to eliminate the arbitrary constants A, B and c

So, differentiating both sides of (i) with respect to x, we get

dy/dx = d/dx {A sinx + B cos (x + c)}

⇒ dy/dx = d/dx (A sinx) + d/dx {B cos (x + c)}

⇒ dy/dx = A cosx - B sin (x + c) ...(ii)

Here,

d/dx (sinx) = cosx

d/dx (cosx) = - sinx

d/dx {k f(x)} = k d/dx {f(x)}, where k is any constant

d/dx {sin (mx + c)} = m cos (mx + c)

d/dx {cos (mx + c)} = - m sin (mx + c)

d/dx (u + v) = du/dx + dv/dx, where u and v are functions of x

Again, differentiating both sides of (ii) with respect to x, we get

d/dx (dy/dx) = d/dx {A cosx - B sin (x + c)}

⇒ d²y/dx² = d/dx (A cosx) - d/dx {B sin (x + c)}

= - A sinx - B cos (x + c)

= - {A sinx + B cos (x + c)}

= - y, using (i) no. equation

⇒ d²y/dx² + y = 0 ...(iii)

which is the required differential equation

The order of the differential equation (iii) is 2 because the highest order of the derivative term d²y/dx² is 2

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