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Differential Equations - Integral Curves and ODEs

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https://brainmass.com/math/integrals/differential-equations-integral-curves-and-odes-508527

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We use various methods to solve several ordinary differential equations.

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Vector Calculus and Linear Algebra

Question 4
Find the work done by the force
F (x,y,z) = -x^2y^3 i + 4j + xk
on moving charged electric particle along the path given by the equation
r (t) = 2cos t i + 2sintj + 4k,
where the parameter t varies from pi/4 to 7pi/4.

Question 5
Displacement of the spring system with friction is described by the differential equation
m d^2y / dt ^ 2 + c dy/dt + ky = 0,
where y is the displacement, m the mass, c the friction coefficient, and k the spring constant (all quantities are non-dimensional). Consider the case c^2 = 4mk (called critical)
a) Find the general solution y(t)
b) Find and sketch the solution satisfying the initial conditions y(0) = 2, y'(0) = -7

Question 7
Consider the vectors
a = (0,3,-2,1,4); b = (5,2,1,0,-1); c = (7, -3,6,21,0)

a) Find the length of the vector v = 2a - b;
b) Are any of the given three vectors parallel or orthogonal? Indicate which (if any)

See attachment for remaining questions and proper notation.

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