Differential Equations Answers

Suppose 𝐴 is real 3 × 3 matrix that has the following eigenvalues and eigenvectors: −2, ( 1 1 1 ) , 1 + 𝑖, ( 1 − 𝑖 2 1 ) , 1 − 𝑖, ( 1 + 𝑖 2 1 ). Find a fundamental set of real valued solutions to 𝐱 ′ = 𝐴𝐱.

Need an explanation as to how they got the following answer:

Answer: The first eigenvalue/eigenvector pair gives the solution 𝐱𝟏 (𝑡) = ( 1 1 1 ) 𝑒^ −2𝑡 . The second eigenvalue/eigenvector pair gives the two solutions: 𝐱𝟐 (𝑡) = ( cos(𝑡) + sin(𝑡) 2 cos(𝑡) cos(𝑡) )𝑒^ 𝑡 , 𝐱𝟑 (𝑡) = ( − cos(𝑡) + sin(𝑡) 2 sin(𝑡) sin(𝑡) )𝑒^ t

A moving body is opposed by a force proportional to the displacement and by a resistance

proportional to the square of velocity. Prove that the velocity is given by

V² = ae − cx/b + /(cab²)

Hint: Equation of motion is mV(dV/dx) = −K₁c − K₂V²

A moving body is opposed by a force proportional to the displacement and by a resistance

proportional to the square of velocity. Prove that the velocity is given by

V²=ae-cx/b+c/(ab²)

Hint: Equation is mv(dv/dx)=(-k₁x-k₂v²)

Solve the homogeneous total differential equation y(y+z)dx +x(x-z)dy + x(x+y)dz=0

(6x+yz)dx+(xz-2y)dy+(xy+2z)dz=0

Find Integrating Factor and then solve the given Differential Equation : (x ^ 2 + y ^ 2 + x) * dx + (x ^ 2 * y - x ^ 3 * y ^ 2) * dy = 0 ?

Find the isogonal trajectories of the family of curves y(x-c)=1 if θ=45

(y^2+yz)dx+(z^2+zx)dy+(y^2-xy)dz=0