4. dy dt -2y+√(t− 3) + 2H(t — 4), y(0) = 1.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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How would I solve number 4 using the solution formula?
Transcribed Image Text:4. \(\frac{dy}{dt} = -2y + \delta(t-3) + 2H(t-4), \, y(0) = 1.\)
This equation is a first-order linear differential equation involving:
- \(\frac{dy}{dt}\): The derivative of \(y\) with respect to \(t\).
- \(-2y\): A term representing exponential decay with a rate proportional to the current value of \(y\).
- \(\delta(t-3)\): The Dirac delta function centered at \(t=3\), representing an impulse or spike at this time.
- \(2H(t-4)\): Twice the Heaviside step function starting at \(t=4\), representing a step input that turns on at \(t=4\).
- \(y(0) = 1\): The initial condition indicating that the value of \(y\) at \(t=0\) is 1.
This type of equation is commonly analyzed in control systems, signal processing, and other applications involving dynamic systems.
![The image presents the following integral equation:
\[
y(t) = y(0)e^{at} + \int_{0}^{t} e^{a(t-s)} q(s) \, ds
\]
### Explanation:
- **\( y(t) \):** This represents the function \( y \) evaluated at time \( t \).
- **\( y(0) \):** This indicates the initial value of the function \( y \) at time \( 0 \).
- **\( e^{at} \):** This is the exponential function with base \( e \) (Euler's number), raised to the power of \( at \), where \( a \) and \( t \) are constants or variables.
- **\( \int_{0}^{t} e^{a(t-s)} q(s) \, ds \):** This represents the integral from 0 to \( t \) of the product of \( e^{a(t-s)} \) and \( q(s) \), with respect to \( s \).
This equation is often used in solving linear differential equations and represents a solution using the method of integrating factors. The first term accounts for the natural exponential growth or decay determined by the initial condition, while the second term represents the effect of a forcing function \( q(s) \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F61f2bd8c-527a-479e-9deb-09c663979694%2Fa132118d-7319-4155-96aa-6fbe3ddfc3f2%2F79lnnen_processed.jpeg&w=3840&q=75)
Transcribed Image Text:The image presents the following integral equation:
\[
y(t) = y(0)e^{at} + \int_{0}^{t} e^{a(t-s)} q(s) \, ds
\]
### Explanation:
- **\( y(t) \):** This represents the function \( y \) evaluated at time \( t \).
- **\( y(0) \):** This indicates the initial value of the function \( y \) at time \( 0 \).
- **\( e^{at} \):** This is the exponential function with base \( e \) (Euler's number), raised to the power of \( at \), where \( a \) and \( t \) are constants or variables.
- **\( \int_{0}^{t} e^{a(t-s)} q(s) \, ds \):** This represents the integral from 0 to \( t \) of the product of \( e^{a(t-s)} \) and \( q(s) \), with respect to \( s \).
This equation is often used in solving linear differential equations and represents a solution using the method of integrating factors. The first term accounts for the natural exponential growth or decay determined by the initial condition, while the second term represents the effect of a forcing function \( q(s) \).
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