基礎應用數學小考

1.Find a constant α such that (x+y)^α is an integrating factor of the equation
(4x²+2xy+6y)dx+(2x²+9y+3x)dy=0
What is the solution of this equation?

2.Solve the following equations:
xdy/dx+y²=1

3.Show that (3x²+ycosx)dx+(sinx-4y³)dy=0 is an exact differential equation and find its general solution.

4.Find the particular integral of (2D²+5D+7)y=3e^2t.

假設 particular integral 形式為 y_p = C e^2t，其中 C 是待定常數。將 y_p 代入，則原式為

 2 C (2)² e^2t + 5 C (2) e^2t + 7 C e^2t = 3 e^2t

( 8 + 10 + 7 ) C = 3

C = 3/25

y_p(t) = (3/25) e^2t

5.Solve:
The equation (D²-D-12)y=0 with the boundary conditions y=0, Dy = 3 when t = 0