Q5Mathematics II

Question

Q.5. The acceleration of a particle at any time $t > 0$ is given by $a = 12 cos 2 t i - 8 sin 2 t j + 16 t k$ . If velocity $v$ and displacement $r$ are zero at $t = 0$ , find $v$ at any time.

Answer 1

Given the second-order linear differential equation with constant coefficients: $(D^{2} - 3 D + 2) y = e^{x}$ .

1. Complementary Function (C.F.)

The fundamental auxiliary equation is exactly $m^{2} - 3 m + 2 = 0$ . Factoring this yields strictly $(m - 1) (m - 2) = 0$ , giving distinct roots $m = 1, 2$ .

C.F. = C_{1} e^{1 x} + C_{2} e^{2 x}

2. Particular Integral (P.I.)

P.I. = \frac{1}{D ^{2} - 3 D + 2} e^{x}

Substituting $D = 1$ strictly makes the denominator zero ( $1 - 3 + 2 = 0$ ), causing the standard rule to fail. According to the strict mathematical failure rule, we must explicitly multiply by $x$ and differentiate the denominator entirely with respect to $D$ :

P.I. = x \frac{1}{2 D - 3} e^{x}

Now substituting $D = 1$ strictly into the new denominator:

P.I. = x \frac{1}{2 ( 1 ) - 3} e^{x} = x \frac{1}{- 1} e^{x} = - x e^{x}

3. Complete Solution

y = C.F. + P.I. = C_{1} e^{x} + C_{2} e^{2 x} - x e^{x}

Q5Mathematics II

Question

Answer

1. Complementary Function (C.F.)

2. Particular Integral (P.I.)

3. Complete Solution