Let y(t) be the solution to the initial-value problem y' + ay = f(t), y(0) = yo, where a and yo are con- stants. Verify that and show that L[y] = y(t) = yoe at L[f] yo + s + a s+a' +6² Le 0 e-a(t-w) f(w)dw.

Q: Find y as a function of tif y(0) = 2, y'(0) = 3. y(t) = 0 16y" + 104y' + 169y = 0,

A:

Q: 6. For what values of y(0) will the solution of y' = y(y – 2),x > 0 a) be a constant function b) be…

A: First of all we draw the direction field of the differential equation y' = y(y-2)

Q: Find the linear approximation to f(x)=x2+1x−2 near a=3.

A: Given: f(x) = x2 +1x − 2; a = 3

Q: Kari is purchasing a home for $260,000. The down payment is 25% and the balance will be financed…

A:

Q: 1. Find the total derivative dz/dy, given (a) z = f(x, y) = 5x + xy - y2, where x = g(y) = 3y² (b)…

A:

Q: Find the derivative of f(x, y) = xey + cosxy at the point (2,0) in the ection.

A: Hi! Thank you for the question, As per the honor code, we are allowed to answer one question at a…

Q: 3. Find the derivative of the function f(x, y, z) = ln (r² + y³ + 4z +5) in the direction of the…

A:

Q: Suppose u and v are differentiable functions of x. Use the given values of the functions and their…

A: Since you have posted a question with multiple sub-parts, we will solve first three subparts for…

Q: Is ƒ(x, y) = x2 - xy + y2 - 3 more sensitive to changes in x or to changes in y when it is near the…

A: Partial derivative of f w.r.t a variable is the indicator of the sensitivity of f towards that…

Q: If G(2) = S. 4t° -2t J-4x TeA _t² +3 dt, then G(x ) = 0. ett True False

A:

Q: . Let r(t) = (t², 1 t, 4t). Calculate the derivative of r(t) · a(t) at t = 6, assuming that a(6) =…

A:

Q: 3. Find and classify any potential extrema of the function (i.e. first and second derivative test)…

A:

Q: Find y as a function of tif y(t) T y" + 16y + 80y = 0, y(0) = 5, y'(0) = 2.

A:

Q: 2z If f(x) = , V1+t° dt, then f'(x) : = 1 Select one: a. -2x/1-8r3 ****** b. -2x/1+ x C. -2/1 – 8x O…

A: Recall the Leibniz rule:

Q: Find y as a function of t if y(0)= 4, y'′(0) = 1. y(t) = 25y" + 30y' +45y = 0,

A: given

Q: (a) F(s) = 2s (s¹ +24s² +144)

A: Since you have asked multiple questions here, we will solve the first question for you. To get the…

Q: Let h(æ, y) = f(x)g(y) + (9(y))² and suppose that f(1) = 5, f'(1) = 3, g(2) = 8, g'(2) = 10. Then…

A: Given function is - h(x,y) = f(x) g(y) + g(y)2             ............................(1)We have to…

Q: Let r(t) = (t2, 1 - t, 4t). Calculate the derivative of r(t) a(t) at t = 4, assuming that a(4) = (9,…

A:

Q: Compute the derivative for r(t) = d dt f(t) = (Use symbolic notation and fractions where needed.)…

A:

Q: Let r(x) = f(g(h(x))), where h(1) = 2, g(2) = 3, h'(1) = 4, g'(2) = 3, and f'(3) 7. Find r'(1) %3D…

A: Recall: The derivative of a composition is the product of the derivative of the outer function (…

Q: Let u(t)=6t3i+t2−4j−5k and v(t)=eti+9e−tj−e9tk. Compute the derivative of the following function.…

A: solve the following

Q: Find the first partial derivative of w = In(1 + )at (2,0, – 1).

A:

Q: (3) Consider the derivative: 2xydx + (x² + 2y)dy. Evaluate whether this is a total derivative or an…

A: We will check whether the given function is total derivative or not. Consider dz = (2xy)dx + (x 2 +…

Q: 2 dz dz dz ds dt dz dy

A:

Q: say f(3)=5, f'(3)=-2 and let g(x) =xf(x/2) find the linearization of g(x) at x=6

A:

Q: If f(x, y, z) = 2x5 + 3y5 - 8x3y²z², what is the partial derivative of f with respect to x? (A 10х4…

A:

Q: -3x2 F(x) = S*" t² dt %3D

A: TO EVALUATE THE FOLLOWING INTEGRAL F(x) ∫-x3x2 t2 dt FORMULA USED- ∫abxn dx = xn+1n+1 i.e substitue…

Q: Find r(t) given that r'(t) = and satisfies r(0) =

A: Step 1:  Step 2: Step 3: Step 4: