1. Find fr, fy, faz, fyy, fay and fyz of (a) f(r, y) = x³ + 3y? – 2x + 4y – 1 (b) f(x, y) = sin (ry) – cos (ry) I + y (c) f(x, y) = (d) f(x, y) = y²e" + x² In y 2. Find all the second order partial derivatives of the functions below: (a) g(x, y): = xe" + y sinx+ cos y (b) h(x, y) = x In y + y ln x + 1 3. Which order of differentiation will calculate fæy faster: x first or y first? (a) f(x, y) = y+ x²y+4y³ – In(y² + 1) (b) f(x, y) = x ln ry a In

Q: showo haw [co(t)+ 3sine) ]= vioet coslt- antan( 3)

A:

Q: Find dy dx 3) y = cos = cos¹ 2 2x

A: We have to find the derivative of y

Q: Question 9: Calculate r (t). r (t) = (4t, 2e2t + 1, 2 sin(t)) ®F(t) = (4t, 2e2ª – 1,2 sin(t)) O F(t)…

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Q: as a function of x when y =

A:

Q: Let F (4t° + 2) i + sin(2t) j+ e*k Find F' (t), F"(t) and F" (t) (1) F'(t) (2) F"(t) (3) F"(t) =

A:

Q: 6. Find * : y- + 5)ao TK, (b) y = Sin 5x I+cos 5x dx

A: Solving the problem by finding the derivative.

Q: Suppose r' (t) = (cos(t), et, 2t) and r(0) = (1, 0, 3). Find r(t).

A:

Q: Let F = (3t + 2) i + sin(2t) j+ 3ť²k Find F'(t), F"(t) and F"(t) 1) F'(t) = (2) F" (t) 3) F" (t) =

A: Solve for the equation for the derivative

Q: Question 8: The curve C in the ry-plane is parameterized by r (t). Find a parameterization of the…

A: Given:r→'t=4, 4e2t, 2costr→0=0,1,0

Q: : Mark all the ODEs in the list for which the function y(t) = cos(t) is a solution. A) y' = sin(t)…

A:

Q: (b) y = e-u cos(u) dy = -e"sin(u) – e"cos( u) os u

A: The given function,  y=e-u cos(u)

Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.