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Consider a spring-mass system that has a 15 kg mass, a damping coefficient of 45 Spring that, when stretched 0.4 m, imposes a force of 12 N. a.) Explain why the spring constant is 30 N/m. 30 b.) These values lead to the equation, y + 5y + y = 0, assuming no forcing function is present. Solve for the position as a function of time by using an appropriate guess. Be sure to show all work. (You can round to two decimal places, e.g., 94.3453 = 94.35) c.) Now, suppose we incorporate a forcing function of e-2t, such that our equation becomes y" + 3y + 2y = e 2t. We might imagine our particular guess to be of the form ae 2t, however, it at e 2t. With this, apply the Linearity will not work in this case. So, use a guess of the form ypa.t·e Principle to find the general solution to this nonhomogeneous ODE.

Consider a spring-mass system that has a 15 kg mass, a damping coefficient of 45 Spring that, when stretched 0.4 m, imposes a force of 12 N. a.) Explain why the spring constant is 30 N/m. 30 b.) These values lead to the equation, y + 5y + y = 0, assuming no forcing function is present. Solve for the position as a function of time by using an appropriate guess. Be sure to show all work. (You can round to two decimal places, e.g., 94.3453 = 94.35) c.) Now, suppose we incorporate a forcing function of e-2t, such that our equation becomes y" + 3y + 2y = e 2t. We might imagine our particular guess to be of the form ae 2t, however, it at e 2t. With this, apply the Linearity will not work in this case. So, use a guess of the form ypa.t·e Principle to find the general solution to this nonhomogeneous ODE.

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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Consider a spring-mass system that has a 15 kg mass, a damping coefficient of 45 N/m², and a spring that, when stretched 0.4 m, imposes a force of 12 N. a.) Explain why the spring constant is 30 N/m. 45 b.) These values lead to the equation, y + 15y + 3y = 0, assuming no forcing function is present. Solve for the position as a function of time by using an appropriate guess. Be sure to show all work. (You can round to two decimal places, e.g., 94.3453 = 94.35) c.) Now, suppose we incorporate a forcing function of e-2t, such that our equation becomes y" + 3y + 2y = e 2t. We might imagine our particular guess to be of the form ae 2t, however, it will not work in this case. So, use a guess of the form ypat e 2t. With this, apply the Linearity Principle to find the general solution to this nonhomogeneous ODE.
Transcribed Image Text:Consider a spring-mass system that has a 15 kg mass, a damping coefficient of 45 N/m², and a spring that, when stretched 0.4 m, imposes a force of 12 N. a.) Explain why the spring constant is 30 N/m. 45 b.) These values lead to the equation, y + 15y + 3y = 0, assuming no forcing function is present. Solve for the position as a function of time by using an appropriate guess. Be sure to show all work. (You can round to two decimal places, e.g., 94.3453 = 94.35) c.) Now, suppose we incorporate a forcing function of e-2t, such that our equation becomes y" + 3y + 2y = e 2t. We might imagine our particular guess to be of the form ae 2t, however, it will not work in this case. So, use a guess of the form ypat e 2t. With this, apply the Linearity Principle to find the general solution to this nonhomogeneous ODE.
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