Consider a dynamic system described by the following second-order differential equation ÿ(t)+3y(t)y(t)+y(t)=3sin(t) with the initial conditions y(0) = 0, and y(0) = 0. i) Choose suitable state variables and derive the state space system model. Give the corresponding initial conditions. ii) For = f(t, y), y(0) = yo, the 4th -order Runge-Kutta method is dt given by Yi+s Y₁+₁ = Y₁ + − (k₁ + 2k₂ + 2k3 + kä)h + k, = S(1,3;), k₂ = f(t, + = h, y; +=k;h), where k; = f[ t, + ½ h, 3, + ½ k₂h)and k, = f(t, +h‚y; +k;h). Using the 4th order Runge-Kutta method and choosing the step length h = 0.1, find the approximate values of y(t) and y(t) at t = 0.1, and t= 0.2.
Consider a dynamic system described by the following second-order differential equation ÿ(t)+3y(t)y(t)+y(t)=3sin(t) with the initial conditions y(0) = 0, and y(0) = 0. i) Choose suitable state variables and derive the state space system model. Give the corresponding initial conditions. ii) For = f(t, y), y(0) = yo, the 4th -order Runge-Kutta method is dt given by Yi+s Y₁+₁ = Y₁ + − (k₁ + 2k₂ + 2k3 + kä)h + k, = S(1,3;), k₂ = f(t, + = h, y; +=k;h), where k; = f[ t, + ½ h, 3, + ½ k₂h)and k, = f(t, +h‚y; +k;h). Using the 4th order Runge-Kutta method and choosing the step length h = 0.1, find the approximate values of y(t) and y(t) at t = 0.1, and t= 0.2.
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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Question
![Consider a dynamic system described by the following second-order
differential equation
ÿ(t)+3y(t)y(t)+y(t)=3sin(t)
with the initial conditions y(0) = 0, and y(0) = 0.
i)
Choose suitable state variables and derive the state space system
model. Give the corresponding initial conditions.
ii)
iii)
dy
- = f (†, y), y(0) = yº, the 4th -order Runge-Kutta method is
dt
given by
Yis₁ = Y; + − (k₁ + 2k₂ + 2kz+ k₂]h
where k; = S(1,3);), k₂ = f( 1, + ½ h,y), + ½ k;h),
k; = f[ t, + ½ h, y, + — k;h)and k₂ = f(t, +h;y, +k;h).
Using the 4th order Runge-Kutta method and choosing the step
length h = 0.1, find the approximate values of y(t) and y(t) at t =
0.1, and t = 0.2.
Draw a simulation diagram implemented in SIMULINK
For](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fed918edd-c392-4c7e-b6b2-8c8c0582de44%2Fb6e00abe-1832-4610-800e-14c0d819f866%2Fa6p63un_processed.png&w=3840&q=75)
Transcribed Image Text:Consider a dynamic system described by the following second-order
differential equation
ÿ(t)+3y(t)y(t)+y(t)=3sin(t)
with the initial conditions y(0) = 0, and y(0) = 0.
i)
Choose suitable state variables and derive the state space system
model. Give the corresponding initial conditions.
ii)
iii)
dy
- = f (†, y), y(0) = yº, the 4th -order Runge-Kutta method is
dt
given by
Yis₁ = Y; + − (k₁ + 2k₂ + 2kz+ k₂]h
where k; = S(1,3);), k₂ = f( 1, + ½ h,y), + ½ k;h),
k; = f[ t, + ½ h, y, + — k;h)and k₂ = f(t, +h;y, +k;h).
Using the 4th order Runge-Kutta method and choosing the step
length h = 0.1, find the approximate values of y(t) and y(t) at t =
0.1, and t = 0.2.
Draw a simulation diagram implemented in SIMULINK
For
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