5. Find extremals for the following functionals: (b) J(y) = S. (1 + (y")2) dx, y(0)...
3. Find the extremals of the functionals (1) (ii) [ Voy (1 + grą) de.
Q1: Solve the ODE: f) Vyy' + y3/2-1. y(1) = 0. g) (2x +y)dx(2x+y-1)dy 0. i) dx=xy2e": y(2)=0. j) (1 + x*)dy + (1 + y*)dx = 0; y(1) = V3.
Q1: Solve the ODE: f) Vyy' + y3/2-1. y(1) = 0. g) (2x +y)dx(2x+y-1)dy 0. i) dx=xy2e": y(2)=0. j) (1 + x*)dy + (1 + y*)dx = 0; y(1) = V3.
(1 - -*)dx + (2xy + + + or? y? 2-)dy = 0 y J
Find extremals
b) J(y) oe2(y - y2)d, y(0)=1, y(3) free c) J(y) Jo2+'y )d, y(0) = , y(1) free. d) J(y) 22y2 - y2)d, y(1) 1, y(e) free. (1) X= B (ii) y is a geodesic; (iii) Xp is not zero and is parallel along 6.5 Let y be a curve with Y(0) = P, T(1) = Q. If Xe T,M, let X be the parallel translate ofX along y. Define y: T,M --- TqM by y*= X(1) (a) Prove...
13. Integrate: a. j«x+278)dx 0 b. (dx х c. dx 9+ x d . xdx? +2 dx 2x+1 хр '(x’+x+3) f. I sin (2x) dx g. cos (3x) dx h. ſ(cos(2x)+ + secº (x))dx i. [V2x+1 dx j. S x(x² + 1) dx k. | xe m. [sec? (10x) dx 16 n. .si dx 1+x 0. 16x 1 + x dx 5 P. STA dx 9. [sec xV1 + tan x dx 14. Given f(x)=5e* - 4 and f(0) =...
Question 2 (Learning Outcome 2) 0 S (*x+3) dx S A) Evaluate the following integrals. 4x+7 2x+5) 5x2–2x+3 (ii) dx (x2+1)(x-1) x2+x+2 (iii) S3x3 –x2+3x+1 dx dx (x+1)V-x-2x In (x) dx (iv) S x2 X+1 (vi) S dx (1+x2) (vii) S dx x(x+Inx) (viii) Stancos x) dx (ix) 30 Sin3 e*(1 + e*)1/2 dx dx 2 sin x cos x (x) S B) Find the length of an arc of the curve y =*+ *from x = 1 to x...
Consider the following system:
dx/dt=y(x^2+y^2-1)
dy/dt= -x(x^2 +y^2-1)
Find the equilibrium solution.
13. Consider the following system dx dy (e) Find the equilibrium solutions (0 Use Maple to sketch a phase portrait (me to understand the qualitative behavior of
13. Consider the following system dx dy (e) Find the equilibrium solutions (0 Use Maple to sketch a phase portrait (me to understand the qualitative behavior of
Compute the
followingelementaryintegrals.(a)dxxxxx3233/2[ 5
](b)dxxx222[5](c)dxxsin3/0
Question [15] Compute the following elementary integrals. 3x dx 3e-*-2 ex+1 [5] x3 sec?(x4 + 2) dx [4] (b) S (0) Lix - 1dx [6] Question 2 [20] Evaluate the following indefinite integrals. x2 dx [6] (b) f(x+1) tan-+ x dx [6] 10x (c) S x + 3)(x2 + 4x + 13) dx [8] Question 3 [15] (a) Find the area of the region bounded by the parabola x = y2 – 5 and the...
Find the general solution of the following diff eq: xdy/dx-3y-2x^5=0 y(1)=4
7. Solve the following differential equations. dy 2 y= 5x, x>0. + a) dx dx 1+2x 4e', t>0 b) t dt
7. Solve the following differential equations. dy 2 y= 5x, x>0. + a) dx dx 1+2x 4e', t>0 b) t dt