1) Consider the differential equationdxa' = 2x + cos? t, x(0) = ro-dti) Solve the differential equation (1) and explain the domain of the solution x(t).ii) Sketch the graph of the solution obtained in item i). Choose your initial condition 2(0) = 2o.2) Consider the differential equationdxr' = -3x + cost + cos 2t, r(0) = ro.dti) Solve the differential equation (2) and explain the domain of the solution x(t)ii) Sketch the graph of the solution obtained in item i). Choose your initial condition (0) = ro.3) Consider the differential equationa' = (cost + sin 2t)x, x(0) = xo.i)Solve the above equation and sketch your solutions with the following initial conditions: (0)x(0) = 1 e r(0) = -1.ii) Sketch the graph of the function X(t) =¤(t)dt, with r(0) = 0 in each of the situations analyzedn in item i). calculate lim-40 X (t).= 0,4)Consider the differential equation a' = (- tan t + sin 2t)æ, æ(0) = xo.i) Solve the above equation and sketch your solutions with the initial conditionsx(2m) = 1. Plot the graph of x(t) in each of the 3 situations described.Note: You can use computer resources. Note that has is not defined across the entire line.r(0) = 0, x(7) = 1 and5) Consider the differential equation of separate variablesdy= / = Os 20, y(0) = y0dæi) Solve equation (3) and calculate the domain of the solution y(x).ii) Plot the graph of solution y.

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Answered: 1) Consider the differential equation…

1) Consider the differential equation dr =r' = 2x + cos²t, r(0) = ro- dt i) Solve the differential equation (1) and explain the domain of the solution x(t).

Advanced Engineering Mathematics

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ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.

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1) Consider the differential equation
dx
a' = 2x + cos? t, x(0) = ro-
dt
i) Solve the differential equation (1) and explain the domain of the solution x(t).
ii) Sketch the graph of the solution obtained in item i). Choose your initial condition 2(0) = 2o.
2) Consider the differential equation
dx
r' = -3x + cost + cos 2t, r(0) = ro.
dt
i) Solve the differential equation (2) and explain the domain of the solution x(t)
ii) Sketch the graph of the solution obtained in item i). Choose your initial condition (0) = ro.
3) Consider the differential equationa' = (cost + sin 2t)x, x(0) = xo.
i)Solve the above equation and sketch your solutions with the following initial conditions: (0)
x(0) = 1 e r(0) = -1.
ii) Sketch the graph of the function X(t) =¤(t)dt, with r(0) = 0 in each of the situations analyzed
n in item i). calculate lim-40 X (t).
= 0,
4)Consider the differential equation a' = (- tan t + sin 2t)æ, æ(0) = xo.
i) Solve the above equation and sketch your solutions with the initial conditions
x(2m) = 1. Plot the graph of x(t) in each of the 3 situations described.
Note: You can use computer resources. Note that has is not defined across the entire line.
r(0) = 0, x(7) = 1 and
5) Consider the differential equation of separate variables
dy
= / = Os 20, y(0) = y0
dæ
i) Solve equation (3) and calculate the domain of the solution y(x).
ii) Plot the graph of solution y.

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9780470458365

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