Math  /  Calculus

Question2. The mathematical model: d2ydt2+7dydt+y=sin2t\frac{d^{2} y}{d t^{2}}+7 \frac{d y}{d t}+y=\sin 2 t is a. linear, time-invariant model b. linear, time-varying model c. Nonlinear, time-varying model d. Nonlinear, time-invariant model

Studdy Solution

STEP 1

What is this asking? Is this differential equation linear, and do its coefficients change over time? Watch out! Don't mix up *time-invariant* with *nonlinear*.
A time-invariant equation has constant coefficients, while a linear equation has the dependent variable and its derivatives appearing only to the first power.

STEP 2

1. Check for linearity
2. Check for time-invariance

STEP 3

Let's peep this equation: d2ydt2+7dydt+y=sin2t\frac{d^{2} y}{d t^{2}}+7 \frac{d y}{d t}+y=\sin 2 t.
A **linear differential equation** is one where the dependent variable (yy in this case) and its derivatives appear only to the **first power**.
They can be multiplied by constants, but no funny business like y2y^2, dydt3\frac{dy}{dt}^3, or ydydty \cdot \frac{dy}{dt} is allowed!

STEP 4

Looking at our equation, we see d2ydt2\frac{d^{2} y}{d t^{2}}, which is the second derivative of yy to the first power.
We also see 7dydt7 \frac{d y}{d t}, which is the first derivative of yy to the first power, multiplied by a constant.
And finally, we have plain old yy, which is also to the first power.
So far, so good!

STEP 5

On the right-hand side, we have sin2t\sin 2t.
This doesn't involve yy at all, so it doesn't affect the linearity of the equation.
It's just a function of tt, which is perfectly acceptable.

STEP 6

Since all terms involving yy and its derivatives are to the first power, we can confidently say that this differential equation is **linear**!

STEP 7

A **time-invariant** equation has coefficients that don't change with time.
In our equation, the coefficient of d2ydt2\frac{d^{2} y}{d t^{2}} is **1**, the coefficient of dydt\frac{d y}{d t} is **7**, and the coefficient of yy is **1**.
All these coefficients are constants, they don't have any tts in them.

STEP 8

Since the coefficients are constant, the equation is **time-invariant**!
The behavior of the system described by this equation doesn't change over time.

STEP 9

The equation d2ydt2+7dydt+y=sin2t\frac{d^{2} y}{d t^{2}}+7 \frac{d y}{d t}+y=\sin 2 t is a **linear, time-invariant model**, so the answer is (a).

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