Chapter 1: Q7RP (page 1)

Decide whether the statement made is True or False. The function $x (t) = t^{- 3} \sin t + 4 t^{- 3}$ is a solution to $t^{3} \frac{dx}{dt} = \cos t - 3 t^{2} x$ .

Short Answer

Expert verified

The statement is true.

Step by step solution

Finding the derivative of the equation x(t)=t-3sin t+4t-3 with respect to t.

Consider, $x (t) = t^{- 3} \sin t + 4 t^{- 3}$

Then,

$\frac{dx}{dt} = \frac{d}{dt} (t^{- 3} sint + 4 t^{- 3}) = t^{- 3} . \frac{d}{dt} (sint) + sint \frac{d}{dt} (t^{- 3}) + \frac{d}{dt} (4 t^{- 3}) = t^{- 3} \cdot \cos t + \sin t (- 3 t^{- 4}) - 12 t^{- 4}$

Putting the value of dxdt in the LHS of the equation t3dxdt=cos t-3t2x to check if RHS is obtained

Equating Left and Right-hand side,

$LHS = t^{3} \frac{dx}{dt} = t^{3} (t^{- 3} \cdot \cos t + \sin t (- 3 t^{- 4}) - 12 t^{- 4}) = \cos t - 3 t^{- 1} \cdot \sin t - 12 t^{- 1}$

$RHS = \cos t - 3 t^{2} x = cost - 3 t^{2} (t^{- 3} \sin t + 4 t^{- 3}) = cost - 3 t^{- 1} \cdot \sin t - 12 t^{- 1}$

Hence, LHS = RHS

Therefore, this shows that $x (t) = t^{- 3} sint + 4 t^{- 3}$ is a solution to $t^{3} \frac{dx}{dt} = cost - 3 t^{2} x$ .

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Decide whether the statement made is True or False. The function $x (t) = t^{- 3} \sin t + 4 t^{- 3}$ is a solution to $t^{3} \frac{dx}{dt} = \cos t - 3 t^{2} x$ .

Short Answer

Step by step solution

Finding the derivative of the equation x(t)=t-3sin t+4t-3 with respect to t.

Putting the value of dxdt in the LHS of the equation t3dxdt=cos t-3t2x to check if RHS is obtained

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Decide whether the statement made is True or False. The function x(t)=t-3sint+4t-3 is a solution to t3dxdt=cost-3t2x.

Short Answer

Step by step solution

Finding the derivative of the equation x(t)=t-3sin t+4t-3 with respect to t.

Putting the value of dxdt in the LHS of the equation t3dxdt=cos t-3t2x to check if RHS is obtained

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Decision Maths

Logic and Functions

Probability and Statistics

Theoretical and Mathematical Physics

Mechanics Maths

Study anywhere. Anytime. Across all devices.

Decide whether the statement made is True or False. The function $x (t) = t^{- 3} \sin t + 4 t^{- 3}$ is a solution to $t^{3} \frac{dx}{dt} = \cos t - 3 t^{2} x$ .