Question

Find the values of p and q such that

4x² + 12x = 4( x + p²) - q




Subject - Completing the square - Maths

I NEED HELP PLEASE :(

Answer 1

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