Proceed as in this example to solve the given boundary-value problem. a²u au ax? 0 < x < 1, t> 0, at u(0, t) = 1 – e-t, u(1, t) = 1 – e-t, t> 0 u(x, 0) = 0, 0

Q: Suppose a person wants to travel D miles at a constant speed of (10+x) mi/hr, where x could be…

A: Introduction: A linear approximation is a mathematical term that refers to the use of a linear…

Q: 2 (y – k) y' = 4a Find the Derivative - d/dy Since 4a is constant with respect to y, the derivative…

A: (1). The given expression is:2y-ky'=4a

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: Eliminate the paranoter t from the follewing S r(1) = 4+ sin t lu(t) 3- cost

A:

Q: TLP iadade mnts & smt Findf F'(t) =sec(t) (seclE)+ tan (ED), -st < f- FCt) = -<ts f (--7 2.

A: We simplify f'(t) by expanding  To find f(t) we integrate f'(t) C=integrating constant

Q: By using first fundamental theorem of calculus. Solve the problem 1.) Suppose f(x, y) = S*3 If fx…

A:

Q: 3. Let h(x, y) x + y sin²x 2¹+ y² 0, if (x, y) = (0,0) if (x, y) = (0,0) (a) Show that h is…

A:

Q: 15) Consider the vector field F (x, y, 2) = (In y, x, z) and the oriented curve C:r (t) = (t, e', 2…

A:

Q: Evaluate 2æz dV where E = {(x, y, 2) | 2 < * < 3, z < y < 2x, 0 < z< z + 2y}

A:

Q: (b) y'+y=f(t), y(0) = 0, where f(t) = 0 for 0 ≤ t < 1 and f(t) = 5 for t ≥1.

A: The given differential equation is , .Where .We know that a differentiable function is continuous as…

Q: Let u = u(x, t), x, t ≤ R. Solve Ut + 5ux + u = 0 u(x, 1) = = COS X

A:

Q: y' + E) y = xy? 2

A:

Q: 7) Suppose w=rt+rs' and r= tan-' (2x² +3y), s=x²y' +3y and t = x²y² .Use the chain rule (from…

A: Given w=r2t+rs3r=tan−1(2x2+3y)s=x2y3+3yt=x2y2

Q: Show that the displacement u(x, t) in an elastic string of length t fireed at both ends which is set…

A:

Q: 4 a. Consider the i.v.p x' = t^(2) + cos(x), x(0) = 0. Verify that the hypothesis of Cauchy Picard…

A:

Q: Solve the initial value problem below using the method of Laplace transforms. y" - 4y' - 12y = 0,…

A: Given equation is: y"-4y'-12y=0 ,y(0)=1,y'(0)=38 We have to find the value of y(x)

Q: f(r, y, z) = sec='(9x + 3yz) f. = fy = f; =

A:

Q: find dy of X sin 2y =YCO52X

A: Given: x sin 2y=y cos 2x To Find: To find dydx.

Q: 4 a. Consider the i.v.p x' = t^(2) + cos(x), x(0) = 0. Verify that the hypothesis of Cauchy Picard…

A: a. Verifying Conditions of Cauchy-Picard Theorem  b. Estimating the Interval of Existence

Q: 1 Find the points of discontinuity of y = where и +и - 2 1 u = * - 1

A:

Q: 2t k, then the distance traveled over 0Sts In 3 is:

A:

Q: tiven the IVP y" +2y" - y' - 2y = 0, y(0) = 1, y'(0) = 2, y"(0) = 0 use the existence nd uniqueness…

A:

Q: Find whether the function F-(* - y')i + (v² – 3x)j + (=* - xy)k is irrotational.

A: A vector field F is called irrotational if curl (F) = 0.

Q: Show that y = x° + 3x +1 Satisfies y" + xy" – 2y' = 0.

A:

Q: Determine the interval of existence and uniqueness without solving

A:

Q: Find o r(t)dt if r(t) = (cos t, sin t, 3t²). %3D O A.O B. (0, 2, ³) OC (0,0, x³) O D. (0, 2, 7) O E.…

A: We have to choose the correct answer from the given option