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d²y ) Solve the differential equation +y = dx² y'(0) = 0. sin(x) + cos(x) with y(0) = 1 and

d²y ) Solve the differential equation +y = dx² y'(0) = 0. sin(x) + cos(x) with y(0) = 1 and

Advanced Engineering Mathematics
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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. (a) Solve the differential equation d²y dx² +y = sin(x) + cos(x) with y(0) = 1 and y'(0) = 0. (b) Consider an object of mass m, dropped from some initial height and falling down towards Earth's surface with velocity v(t) at time t. Suppose we choose to model the force of air resistance as being proportional to the square of the instantaneous speed. (i) If we choose "up" to be the positive y direction and set ß> 0 to be a con- stant of proportionality, briefly explain why the most appropriate differential equation to model this setup is dv = -mg+Bv² with v(0) = 0, dt rather than dv m- = -mg - Bv² with v(0) = 0. dt (ii) After setting m = 1, B 1, B = 10 and g = 10, solve this differential equation for the velocity of the object at time t. (iii) Determine the time at which the object reaches 90% of its terminal velocity. 3 (c) Evaluate √3x - x² dx. m-
Transcribed Image Text:. (a) Solve the differential equation d²y dx² +y = sin(x) + cos(x) with y(0) = 1 and y'(0) = 0. (b) Consider an object of mass m, dropped from some initial height and falling down towards Earth's surface with velocity v(t) at time t. Suppose we choose to model the force of air resistance as being proportional to the square of the instantaneous speed. (i) If we choose "up" to be the positive y direction and set ß> 0 to be a con- stant of proportionality, briefly explain why the most appropriate differential equation to model this setup is dv = -mg+Bv² with v(0) = 0, dt rather than dv m- = -mg - Bv² with v(0) = 0. dt (ii) After setting m = 1, B 1, B = 10 and g = 10, solve this differential equation for the velocity of the object at time t. (iii) Determine the time at which the object reaches 90% of its terminal velocity. 3 (c) Evaluate √3x - x² dx. m-
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