Use Laplace transform method to solve the following ordianry differential equa- tion: y" (t) + 4y' (t) + 5y(t) = e'; Initial conditions are y(0) = 1, y'(0) = 2.

Q: Let r(t) = 5 cos(t) and y(t) = 3 sin(t) All answers should be decimals rounded to 2 decimal places.…

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Q: Suppose h(x) = cos(tť² )dt Calculate h' (x).

A: Given: hx=∫0x3cosπt2dt.

Q: 1. Find the function y(t) satisfying y' = ev-t(1+t?) sec y with y(0) = 0.

A: Claim: solution for with y(0)=0. Integrating both sides we get, Take u=e-y then du=-e-y  and…

Q: Find g'(x) if g(x) = t² cos 6t dt %3D 2

A: We can apply fundamental theorem of calculus here.  The second fundamental theorem of calculus…

Q: Given h(x) = f(x) · g(x) f(x) • j(x) = %3D g(x) - 5, the equation of the tangent line of f is y – 7…

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Q: Problem 2 (Completing the proof of Proposition 1) Let V be a vector space. Prove that the following…

A: Steps of solution Part 1) To prove that for every scalar c∈R, the equation c⋅0v​=0v​​ holds in a…

Q: 2. Let F(x) = x s sin (t2) dt. Find F'(x). (a) sin (x²) dx (b) sin (x²) (c) sin (x²) + C (d) 2x sin…

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Q: Q4 B) Find y' for the functions cos(xy) + 3*+cos ly+ In(xy) + log-7 dy

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Q: a) Suppose that f(4)=3 and f'(4)=-5. Find g'(4) where g(x)=Vx f(x) b) Let y=cosx-sin.x , show that…

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Q: Use a linear approximation (or differentials) to estimate the given number to the nearest hundredth.…

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Q: Find G(1), G'(0), and G'(π/4), where G(x) =

A: Here we have;  G(x)=∫1xtant dt

Q: (a) y=In, X-4 2x+5

A: According to bartleby guidelines we supposed to do the first one of multiple questions kindly repost…

Q: F(x) = J e2f sin² (3t)dt. Then F'(x) =

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Q: If f(x) = x^2 sin(3x), find the differential dy

A: Consider the given: fx=x2sin3xy=x2sin3x Differentiate w.r.t. "x", dydx=ddxx2sin3xApply the Product…

Q: dy Find for y= csc (x)+ dx cos (x'+1). Do not simplify your answer. dy dx Given the equation x' + y'…

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Q: (a) (b) y' = 2y + t²e²t ty' + 2y = sin(t)

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Q: 8) Let z = y-x². Use differentials to estimate dz when x = 1,y = 5, dx = 0.03, and dy 0.1. !! %3D

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Q: For the Cauchy-Euler equation x2y''+xy'+y=x2 if you substitute z=ln(x), y'=(1/x)(dy/dz), and…

A: The Cauchy-Euler equation x2y''+xy'+y=x2. Find the value of a+b+c=?

Q: Consider the following: 7 cos(x) sin(y) = 8 (a) Differentiate with respect to x (y is a function of…

A: Given: 7cosxsiny=8 We know that product rule of differentiation is defined as  fxgx'=f'xgx+fxg'(x)