Find the solution in a vector form of the given system find

Question

Assignment

I. Solve the Bernoulli's differential equation.

dy/ dx = (-3x² + 2y²) / 4xy

II. Find the particular solution of the homogeneous differential equation with the given initial condition.

xdy = (y + 3√xy)dx,     y(1) = 0

III. Determine whether the given differential equations are exact. If it is exact, then find its general solution

1. (x³ + 8y - 3x) dy = 3y (x² - 1) dx

2. (2xy - tan y) dx +  (x² - x sec² y) dy = 0

IV. Find explicit (if convenient) general solution of the given differential equation

(4 - x²) dy - (1 +y)² dx = 0

V. Solve the linear system of equations using Gaussian elimination.

2x₁ + 3x₂ - x₃ + 3 x₄ = 11
x₁ + x₂ + x₃ - x₄ = -1
x₁ + x₂ + x₃ - x₄ = -4

VI. Find the solution in a vector form of the given system.

x₁ + 3x₄ - x₅ = 0
x₂ - 2x₄ + 6x₅ = 0
x₃ + x₄ - 8x₅ = 0