3. Let y" +2y' - 3y = f(x). Find the solution in the cases (a) f(x)=0; (b) f(x) 6x; (c) f(x) = 4 , y(0)-0, y'(0) - 1.
af af. and 1. Fоr f(x, y) = ?у? + 5xy?, find Әх ду 1 Әf af 2. Fоr f (x, y) = ln(x? + 3у?), find and Әr ду N 1
5. Let f R2 ->R2 be the function given by f(x, y) (х + у, х — у). (i) Prove that f is linear as a function from R2 to R2. (ii) Calculatee the matrix of f. (iii) Prove that f is a one-to-one function whose range is R2. Deduce that f has an inverse function and calculate it. (iv) If C is the square in R2 given by C = [0,1] x [0, 1], find the set f(C), illustrating...
Please find and classify all the critical points for Q19 and Q20 5-20 Find the local maximum and minimum values and saddle point(s) of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function 5. f(x, y) xy y + y 6. f(x, y)-xy 2x 2y x-y 7. f(x, y) x-y)1 - xy) 8. f(x, y)y(e- ) 9. f(x, y)-x y* + 2xy 10. f(x,...
3. Given f(x,y)= sin?(2x+3y?).e***; (a) Find f (x,y). (b) Find f (x,y).
Problem #2: Use the given graphs to sketch the parametric curve x =f(0, y=g(1). х=f(t) y=g(t) A m KA (А) (В) о (D) у х 1 0 2 х E) (F) ТО -1 о 2 (Н) 2 -2 2 х Problem #2: Select
(10 points) For the differential equation y(6) - 2y (5) – 3y(4) + 2y(3) + 10y" – 8y = 0. Find the fundamental solution set to the DE if the characteristic equation in factored form is given by (r – 2) (r2 + 2r + 2) (r - 1) (r + 1) = 0
Find the solution of the given IVP y" + 3y' + 2y = uz(t); y(0) = 0, y'(0) = 1 a. y = et-e-t + uz(t) [] + e-(6+2) +22(6+2) b. y = ef +e-t+uz(t)ſ - e-(6-2) + şe-26-2)] + uz(t) - e-(1-2) 3e=2(-2)] e + C. y = e-t-e-27 d. None of these
In Exercises 71-74, find a function z = f(x,y) whose partial deriva- tives are as given, or explain why this is impossible. af af af 2y ar (x + y)2, 2r 73. (x + y)2 In Exercises 71-74, find a function z = f(x,y) whose partial deriva- tives are as given, or explain why this is impossible. af af af 2y ar (x + y)2, 2r 73. (x + y)2
15.8 a. Use Stokes' Theorem to evaluate fF.dr where F(x,y,z) = (32-2y)i + (4x – 3y)j + (z +2y)k and C is the boundary of the triangle joining the points (1, 0, 0), (0, 1, 0), and (0, 0, 1). b. Find F.dr where F = 2zi - xj + 3y2k and S is the portion of the plane 3x + 3y + 2z = 6 in the first octant and C is its boundary.