Find the values of p and q such that
4x² + 12x = 4( x + p²) - q
Subject - Completing the square - Maths
I NEED HELP PLEASE :(
Answers
Answer:
Step-by-step explanation:
The general form of the first order linear differential equation is as follows
dy / dx + P(x) y = Q(x)
where P(x) and Q(x) are functions of x.
If we multiply all terms in the differential equation given above by an unknown function u(x), the equation becomes
u(x) dy / dx + u(x) P(x) y = u(x) Q(x)
The left hand side in the above equation has a term u dy / dx, we might think of writing the whole left hand side of the equation as d (u y ) / dx. Using the product rule of derivatives we obtain
d (u y ) / dx = y du / dx + u dy / dx
For y du / dx + u dy / dx and u(x) dy / dx + u(x) P(x) y to be equal, we need to have
du / dx = u(x) P(x)
Which may be written as
(1/u) du / dx = P(x)
Integrate both sides to obtain
ln(u) = ò P(x) dx
Solve the above for u to obtain
u(x) = eò P(x) dx
u(x) is called the integrating factor. A solution for the unknown function u has been found. This will help in solving the differential equations.
d(uy) / dx = u(x) Q(x)
Integrate both sides to obtain
u(x) y = ò u(x) Q(x) dx
Finally solve for y to obtain
y = ( 1 / u(x) ) ò u(x) Q(x) dx