Answer all the questions in the please Solve [25 marks] 1. y' - y = 2x² cos(x), y(3) =1 [6 marks] 2. y' – 4y = 1 – e¥y', y(0) = 0 [6 marks] 3. yy' = 2y2 In(x) + 2xy2 – In(x) – - X [6 marks] 4. xy' + y = vxy, y(4) = = 1 [7 marks] End of Assessment
QUESTION 3 dy dx E-2xy2 and y=3 when x = 1. What is the value ofy when x = 3? 3 23 1 3 0 o w celu 28
Solve the following: 1. x*y'-2*y-2*x^2*y 2. y xty/(x-5) 3. y'y/x, y(1)-2 4. yy+2*exp(2*x), y(0)=3 5. (1+x)*y+ysin(x), y(-pi/2)=0 1. x*y'-2*y-2*x^2*y 2. y xty/(x-5) 3. y'y/x, y(1)-2 4. yy+2*exp(2*x), y(0)=3 5. (1+x)*y+ysin(x), y(-pi/2)=0
Find the value of b for 2xº + 2xy2 + 3x² + 3y2 - if (x, y) = (0,0) 2. a) Let f(x, y) = x + y if (x, y) = (0,0) which fis continuous at all the points in R2. if (x, y) + (0 -2) Is f continuous at (0, -2)? 1 2x + xy b) Let f(x, y) = 3x² + (2 + y)? 12 Explain! if (x, y) = (0, -2)
Let F(x,y,z) = <2y2z, 4xyz, 2xy2> be a vector field. (a) Knowing that F is conservative, find a function f such that F = Vf and f(1,2,1)= 8. (b) Using the result of part(a), evaluate the line integral of F along the following curve C from (0, 0, 0) to (3.9, 1.4, 2.6). y2 + x4z3 + 2xy(x3 + y4 + 24)1/3 = K ; K is a constant Answer: Next page
part c Solve the initial value problem yy' + + y with y(4) - 33 a. To solve this, we should use the substitution u=x^2+y^2 help (formulas '= 2x+2yi help (formulas) Enter derivatives using prime notation (e.g.. you would enter y' for ). N b . After the substitution from the previous part, we obtain the following linear differential equation in ruu 1/2 sqrt() help. (equations e. The solution to the original initial value problem is described by the following...
1. Solve the following ODEs (a) yy (xy)?e_¥/= (b) ryyy22x2 (e) yyxy)
Solve the following differential equation by separation of variable method: 1-xyy' = (y^2) - yy'
Find a solution 2. yy' = x(y2 + 4).
3. Consider the function f(x,y) = 4 + 2x - 3y - x2 + 2y2 - 3xy. a) (5 pts.) Calculate the partial derivative functions, and use them to calculate the gradient vector evaluated at c = b) (5 pts.) Write down the affine approximation to at the e given in a) /(x) = f(c)+ Vf(e)'(x - c) . Use it to calculate (1.1, 1.1). (Hint: it should be close to f(1.1, 1.1))