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Consider the differential equation dx/dt + 2x = 3. Note that this is a first order linear differential equation, where p(t) and g(t) are both continuous functions. 1. Let u be a differentiable function. Use the product rule to expand (xp)' 2. In the equation (dx/dt) + 2x = 3, rewrite dx/dt as x' 3. Notice that the left-hand side of the equation in #2 looks a lot like the expanded product rule but is missing the function u. So multiply both sides by μ, a function that we will determine shortly. 4. Because, so far, μ is an arbitrary function, we can have u satisfy any differential equation that we want. Use μ' = 2u to rewrite the left-hand side of # 3 to look like #1. 5. Use separation of variables to solve μ' = 2μ. 6. Replace µ in the equation from #3 with your solution from #5. 7. Show that the equation in Box 5 can be rewritten as (xe^(2t))' = 3e^(2t) Hint: Consider #1. 8. Write integrals with respect to t on both sides. Apply the Fundamental Theorem of Calculus. 9. Obtain an explicit solution by isolating x(t).

Consider the differential equation dx/dt + 2x = 3. Note that this is a first order linear differential equation, where p(t) and g(t) are both continuous functions. 1. Let u be a differentiable function. Use the product rule to expand (xp)' 2. In the equation (dx/dt) + 2x = 3, rewrite dx/dt as x' 3. Notice that the left-hand side of the equation in #2 looks a lot like the expanded product rule but is missing the function u. So multiply both sides by μ, a function that we will determine shortly. 4. Because, so far, μ is an arbitrary function, we can have u satisfy any differential equation that we want. Use μ' = 2u to rewrite the left-hand side of # 3 to look like #1. 5. Use separation of variables to solve μ' = 2μ. 6. Replace µ in the equation from #3 with your solution from #5. 7. Show that the equation in Box 5 can be rewritten as (xe^(2t))' = 3e^(2t) Hint: Consider #1. 8. Write integrals with respect to t on both sides. Apply the Fundamental Theorem of Calculus. 9. Obtain an explicit solution by isolating x(t).

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Consider the differential equation dx/dt + 2x = 3. Note that this is a first order linear differential equation, where p(t) and g(t) are both continuous functions. 1. Let u be a differentiable function. Use the product rule to expand (xμ)' 2. In the equation (dx/dt) + 2x = 3, rewrite dx/dt as x' 3. Notice that the left-hand side of the equation in #2 looks a lot like the expanded product rule but is missing the function μ. So multiply both sides by μ, a function that we will determine shortly. 4. Because, so far, μ is an arbitrary function, we can have u satisfy any differential equation that we want. Use µ' = 2u to rewrite the left-hand side of #3 to look like #1. 5. Use separation of variables to solve μ' = 2μ. 6. Replace μ in the equation from #3 with your solution from #5. 7. Show that the equation in Box 5 can be rewritten as (xe^(2t))' = 3e^(2t) Hint: Consider #1. 8. Write integrals with respect to t on both sides. Apply the Fundamental Theorem of Calculus. 9. Obtain an explicit solution by isolating x(t).
Transcribed Image Text:Consider the differential equation dx/dt + 2x = 3. Note that this is a first order linear differential equation, where p(t) and g(t) are both continuous functions. 1. Let u be a differentiable function. Use the product rule to expand (xμ)' 2. In the equation (dx/dt) + 2x = 3, rewrite dx/dt as x' 3. Notice that the left-hand side of the equation in #2 looks a lot like the expanded product rule but is missing the function μ. So multiply both sides by μ, a function that we will determine shortly. 4. Because, so far, μ is an arbitrary function, we can have u satisfy any differential equation that we want. Use µ' = 2u to rewrite the left-hand side of #3 to look like #1. 5. Use separation of variables to solve μ' = 2μ. 6. Replace μ in the equation from #3 with your solution from #5. 7. Show that the equation in Box 5 can be rewritten as (xe^(2t))' = 3e^(2t) Hint: Consider #1. 8. Write integrals with respect to t on both sides. Apply the Fundamental Theorem of Calculus. 9. Obtain an explicit solution by isolating x(t).
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