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Solve the following differential equations. Express the solution of the given initial value problem in terms of convolution integral. a) y"+w²y = g(t), y(0) = 0, y'(0) = 1 3.

Solve the following differential equations. Express the solution of the given initial value problem in terms of convolution integral. a) y"+w²y = g(t), y(0) = 0, y'(0) = 1 3.

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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Solve the following differential equations. Express the solution of the given initial value problem in terms of convolution integral. a) y"+w²y = g(t), y(0) = 0, y'(0) = 1 3. = [te-¹f(t-1)dt, y(0) = y'(0) = 0 b) y" + 2y + y =
Transcribed Image Text:Solve the following differential equations. Express the solution of the given initial value problem in terms of convolution integral. a) y"+w²y = g(t), y(0) = 0, y'(0) = 1 3. = [te-¹f(t-1)dt, y(0) = y'(0) = 0 b) y" + 2y + y =
FORMULAS TO REMEMBER L(eat f(t)) = F(s-a); L¹(F (s-a)) = eªt f(t) where f(t) = ¹(F(s)) (f(t-c)u(t)) = e "L(f(t)}; L-¹ (e-F(s)) = f(t-c)u(t) where f(t) = L-¹ (F(s)) c(f(t) 8 (t-c)) = f(c)e-cs (F(s) G(s)) = f(t)-g(t) f(t) = g(t) = [ f(t)g(t-r)dt = ["g(7)f(t-1)dr L(&(t-c)) = e-s; L(f(t) g(t)) = F(s)G(s); 8
Transcribed Image Text:FORMULAS TO REMEMBER L(eat f(t)) = F(s-a); L¹(F (s-a)) = eªt f(t) where f(t) = ¹(F(s)) (f(t-c)u(t)) = e "L(f(t)}; L-¹ (e-F(s)) = f(t-c)u(t) where f(t) = L-¹ (F(s)) c(f(t) 8 (t-c)) = f(c)e-cs (F(s) G(s)) = f(t)-g(t) f(t) = g(t) = [ f(t)g(t-r)dt = ["g(7)f(t-1)dr L(&(t-c)) = e-s; L(f(t) g(t)) = F(s)G(s); 8
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