Differential Equations Answers

1. If u = f(x,y) be a function of two independent variables x and y, then ∂u/∂y is equal to
a) lim△x→0f(x+△x,y)−f(x,y)△x
b) lim△y→0f(x,y+△y)−f(x,y)△y
c) limy→0f(x,y+△y)−f(x,y)△y
d) lim△y→0f(x+△x,y)−f(x,y)△x
2. Find the total differential of the function
u=x2y−3y

a)2x+(x2−3)dy
b) 2xydx+x2dy
c) 2xydx+(x2−3)dy
d) 2xydx+(x3−2)dy
3. The total differential du of a function u (x, y) = 0 is defined as

a) ∂u∂xdx+∂u∂xdy=0
b) ∂u∂xdy+∂u∂xdy=0
c) ∂u∂xdx+∂u∂ydx=0
d)∂u∂xdx+∂u∂ydy=0

A)Solve the following simultaneous inequalities
8x - 3≤ 1/4 [x + 16]< x + 10

B) Differentiate the following functions
y = x^3+〖6x〗^(-4)-〖4x〗^(1/2)+ 10

y = (〖6x〗^3+〖4x〗^2+3)(〖2x〗^2-〖4x〗^(-2)+5)

Solve 2 dy/dx+y tan⁡x = (4x+5)^2/cos⁡x y^3

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y'''+y''=8x^2 metodo anulador

10 One of the following is not a sufficient conditions for the existence of solution of the first order equation dy/dx = f(x,y), with y(x0)=y0
in a region T defined by |x−x0|<a and |y−y0


i. f is continuous in T

II. f is bounded in T

III. ∂f/∂y is continuous in T

IV. ∂f/∂y is not continuous in T