Question

Show that \(x(t)=7 \cos 3 t-2 \sin 2 t\) is a solution of $$ \frac{\mathrm{d}^{2} x}{\mathrm{~d} t^{2}}+2 x=-49 \cos 3 t+4 \sin 2 t $$

Short answer

Short Answer: The given function \(x(t) = 7 \cos 3 t - 2 \sin 2 t\) is a solution to the given differential equation \(\frac{\mathrm{d}^{2} x}{\mathrm{~d} t^{2}}+2 x=-49 \cos 3 t+4 \sin 2 t\), as proven by finding its second derivative and plugging the function and its second derivative back into the equation, which holds true.

Step-by-step solution

Find the first derivative of \(x(t)\)

To find the first derivative of \(x(t)\) with respect to \(t\), we need to differentiate each term separately. For the first term \(7 \cos 3t\), we need to apply the chain rule, which means differentiating the outside function and the inside function separately. For the second term \(-2 \sin 2t\), apply the same process. First derivative of \(x(t)\) with respect to \(t\): $$ \frac{\mathrm{d} x}{\mathrm{~d} t} = \frac{\mathrm{d}(7 \cos 3t)}{\mathrm{~d} t} - \frac{\mathrm{d}(2 \sin 2t)}{\mathrm{~d} t} $$ Using the chain rule for the first term: $$ \frac{\mathrm{d}(7 \cos 3t)}{\mathrm{~d} t} = 7(-3 \sin 3t) $$ Using the chain rule for the second term: $$ \frac{\mathrm{d}(2 \sin 2t)}{\mathrm{~d} t} = 2(2 \cos 2t) $$ Combining both terms: $$ \frac{\mathrm{d} x}{\mathrm{~d} t} = -21 \sin 3t + 4 \cos 2t $$

Find the second derivative of \(x(t)\)

Now we have to take the derivative again to find the second derivative of \(x(t)\) with respect to \(t\). Apply the same steps as in Step 1. Second derivative of \(x(t)\) with respect to \(t\): $$ \frac{\mathrm{d}^2 x}{\mathrm{~d} t^2} = \frac{\mathrm{d}(-21 \sin 3t + 4 \cos 2t)}{\mathrm{~d} t} $$ Differentiating each term separately using the chain rule: $$ \frac{\mathrm{d}^2 x}{\mathrm{~d} t^2} = -21(3 \cos 3t) - 4(2 \sin 2t) $$ Simplifying the expression: $$ \frac{\mathrm{d}^2 x}{\mathrm{~d} t^2} = -63 \cos 3t - 8 \sin 2t $$

Plug \(x(t)\) and its second derivative into the given differential equation

Now that we have both \(x(t)\) and its second derivative, we can plug them back into the given differential equation and check if the equation holds true. Given differential equation: $$ \frac{\mathrm{d}^{2} x}{\mathrm{~d} t^{2}}+2 x=-49 \cos 3 t+4 \sin 2 t $$ Plugging in the \(x(t)\) function and its second derivative: $$ (-63 \cos 3t - 8 \sin 2t) + 2(7 \cos 3t - 2 \sin 2t) = -49 \cos 3 t+4 \sin 2 t $$ Simplifying the equation: $$ -63 \cos 3t - 8 \sin 2t + 14 \cos 3t - 4 \sin 2t = -49 \cos 3 t+4 \sin 2 t $$ Combining like terms: $$ (-63 + 14) \cos 3t + (-8 - 4) \sin 2t = -49 \cos 3 t + 4 \sin 2 t $$ Simplifying further: $$ -49 \cos 3t + 4 \sin 2t = -49 \cos 3 t + 4 \sin 2 t $$ Since the given differential equation holds true, we can conclude that \(x(t) = 7 \cos 3 t - 2 \sin 2 t\) is indeed a solution of the given differential equation.

Key concepts

Chain Rule

The chain rule is a fundamental calculus tool that allows us to differentiate composite functions. In simpler terms, it's a method to find the rate of change of functions that are nested inside one another.

For example, if you have a function like \(h(x) = f(g(x))\), the rate of change of \(h\) with respect to \(x\) is not straightforward to compute by traditional differentiation rules. This is where the chain rule comes into play telling us that the derivative of \(h\) is the derivative of \(f\) with respect to \(g\) multiplied by the derivative of \(g\) with respect to \(x\). Mathematically, it is expressed as: \[\frac{\mathrm{d}h}{\mathrm{d}x} = \frac{\mathrm{d}f}{\mathrm{d}g} \cdot \frac{\mathrm{d}g}{\mathrm{d}x}\]
In the context of differential equations involving trigonometric functions, the chain rule is often used when the argument of the sinusoidal function (like \(3t\) in \(\cos 3t\)) is not just \(x\) but a function of \(x\) itself. This is crucial for correctly finding the first and second derivatives needed to solve the equations.

First Derivative

The first derivative of a function gives us the slope of the curve at any point, which indicates how the function is changing at that specific instant. It’s the first step in determining the behavior of the function. In practical terms, it answers the question: 'At a given point, is the function increasing or decreasing, and how rapidly?'

To find the first derivative of a function, we utilize differentiation rules, such as the chain rule for composite functions, and standard derivatives like those for trigonometric functions. For instance, the first derivative of \(\sin x\) is \(\cos x\), while the derivative of \(\cos x\) is \(−\sin x\). When a constant is multiplied by the function, you keep the constant and multiply it by the derivative of the function, as was done with the terms \(7\cos 3t\) and \(−2\sin 2t\) in our example.

Second Derivative

The second derivative is the derivative of the derivative. It provides insight into the curvature of the function’s graph, that is, how its slope is changing over time. Conceptually, while the first derivative can tell us 'Is the function speeding up or slowing down?', the second derivative adds another layer, telling us 'Is the rate of speeding up itself getting faster or slower?'

In the context of sinusoidal functions, the second derivative will often show the same types of functions we started with, but possibly flipped and/or with different constants due to the properties of trigonometric functions’ derivatives. For example, taking the second derivative of \(\sin\) will give us \(−\sin\), a reflection of the sinusoidal nature of these functions. In the exercise provided, the calculated second derivative \(−63\cos 3t − 8\sin 2t\) is crucial for verifying the solution to the given differential equation.

Sinusoidal Functions

Sinusoidal functions, which include the basic \(\sin x\) and \(\cos x\), are periodic and wave-like, representing phenomena such as sound waves, light waves, and alternating current electric signals. These functions are characterized by their amplitude, frequency, and phase.

In our differential equation example, \(x(t) = 7 \cos 3t - 2 \sin 2t\) combines two sinusoidal functions. The coefficients 7 and −2 are the amplitudes, affecting how high the waves go, and the numbers 3 and 2 inside the functions represent the frequency, which affects how often the waves repeat within a given interval. Solving differential equations involving sinusoidal functions often involves understanding these characteristics, as they will influence the derivatives and the solutions to the equations. Correctly solving such equations can reveal critical details about oscillatory systems in physics and engineering.