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- Date Submitted: 10/16/2010 7:15 AM
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FURTHER MATHEMATICS T 956/1

CHAPTER 8: DIFFERENTIAL EQUATIONS 8.1: FIRST ORDER LINEAR DIFFERENTIAL EQUATIONS Exact differential equation Knowing that d dv du (uv) = u + v dx dx dx We can recognize that right hand side of this formula when it is occurs in differential equation, and quoted its integrals, uv. Example: dy x +y=e dx The left hand side can be recognized as the derivatives with respect to x of the product of xy. Thus, integrating both sides of this equation with respect to x gives xy = e + A A differential equation of this type, in which part of it is exact derivative of a product, is called an exact differential equation. Example: 1. Find the general solution of the differential equation dy 1 x cos y + 2x sin y = dx x Solution: The left hand side seems to be derivatives with respect to x of x sin y Hence, integrate both sides: 1 x sin y = − + A x x sin y = Ax − 1 is the general solution of the given differential equation. 2. Find y in terms of x if y = 0 and x = 1 2ye dy +y e =e dx

Solution: The left hand side seems to be derivatives with respect to x of y e Hence, integrate both sides: 1 y e = e +A 2 1 A y = e + 2 e

y=±

e A + 2 e

when y = 0 and x = 1 1 A 0 = e+ 2 e A e = e 2 e A= 2 e e + 2 2e

∴y=±

The Integrating Factor Consider a first order differential equation that can be written in the form dy + Fy = G dx where F and G are not the functions of x only. The left hand side is not exact, but supposes that it becomes exact when it is multiplied by I, a function of x: dy I + (y)FI = GI dx Comparing the left hand side with du dv v +u : dx dx we have: du dy v = I, = dx dx dv u = y, = FI dx dv Look → when v = I, = FI dx dI ∴ = FI dx Now this is in a first order linear differential equation with separable variable: 1 dI = F dx I ln I = I = e∫ F dx

Thus, we see that e∫

So provided e∫

is an integrating factor for the expression dy + Fy dx can be found, we have: dy + Fy = G dx dy I + (y)FI = GI dx dy I + (y)FI dx = GI dx dx Iy =...