Find the particular integral of the following differential
equations.(Explain each step clearly)
(a) d2y/dx2 + y = (x + 1) sin x. show that
the answer is yp(x) = − 1/8 [ (2x2 + 4x − 1) cos x − (2x
+ 2) sin x ]
(Hint:In this case, we substitute sin αx or cos αx with
eiαx then use the shift operator. In the case of sin αx
we extract the imaginary part.)
Consider the differential equation
2y^2+10y+12 = t+e^t
Find the complementary function and particular integral. Hence
write down the full general solution
Consider the curve given by the equation y^2 - 2x^2y = 3
a) Find dy/dx.
b) Write an equation for the line tangent to the curve at the
point (1, -1).
c) Find the coordinates of all points on the curve at which the
line tangent to the curve at that point is horizontal. d) Evaluate
d 2y /dx2 at the point (1, -1).