MATH 046 Lecture Notes - Lecture 6: Integrating Factor, Product Rule, Lincoln Near-Earth Asteroid Research

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14 Dec 2016

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Document Summary

Linear(cid:1) first(cid:1) order(cid:1) differential(cid:1) equations (cid:46)ethod(cid:1)of(cid:1)integrating(cid:1)factors(cid:1)to(cid:1)solve(cid:1)linear(cid:1) rst(cid:1)order(cid:1)equation. y + p(x)y = q(x) When(cid:1) p(x)(cid:1)=(cid:1)0,(cid:1) the(cid:1) equation(cid:1) is(cid:1) simply y (x) = q(x). This means that y is the anti derivative of q(x) and we simply have. For(cid:1) the(cid:1) equation the(cid:1) general(cid:1) solution(cid:1) is y = (cid:2) q(x)dx. y (x) = 2x, y = (cid:2) 2xdx = x2 + c. When(cid:1) p(x)(cid:1)=(cid:1)0,(cid:1) the(cid:1) di erential(cid:1) equation(cid:1) is(cid:1) easy(cid:1) to(cid:1) solve. (cid:1) our(cid:1) goal(cid:1) is(cid:1) to(cid:1) use(cid:1) integrating(cid:1) factors(cid:1) to(cid:1) rewrite(cid:1) any(cid:1) linear(cid:1) equation(cid:1) into(cid:1) this(cid:1) simple(cid:1) form. Let(cid:1)us(cid:1)recall(cid:1)the(cid:1)product(cid:1)rule(cid:1)of(cid:1)di erentiation. (cid:1)if(cid:1)we(cid:1)have(cid:1)two(cid:1)functions(cid:1)f (x)(cid:1)and(cid:1)g(x),(cid:1) then(cid:1)the(cid:1)derivative(cid:1)of(cid:1)their(cid:1)product(cid:1)is (cid:1) product(cid:1) rule (f g) (cid:1) =(cid:1)f g(cid:1)+(cid:1)f g . Example(cid:1) (xex) = (x) ex + x(ex) = ex + xex. When solving di erential equation, we frequently apply the product rule where one of the functions is the unknown. In that case, it looks like the following example. Let"s(cid:1) see(cid:1) how(cid:1) to(cid:1) apply(cid:1) the(cid:1) method(cid:1) of(cid:1) integrating(cid:1) factor(cid:1) to(cid:1) solve(cid:1) the(cid:1) following(cid:1) equation.

Related Questions

Follow the six steps below to solve the following: xy' + 6xy =xe^(5x)

Step 1: REWRITE into standard form (y' + P(x)y + Q(x))

a) How does the given equation differ from standard form?

b) What can be done to put the given equation into standardform?

Step 2: IDENTIFY P(x) and Q(x).

a) Where does the standard form indicate P(x) and Q(x) arelocated?

P(x) =

Q(x) =

Step 3: DETERMINE the integrating factor (e^(?P(x) dx)). PlugP(x) as identified in step 2 into the integrating factor forme^(?P(x) dx) . Hint: Simplify e^(?P(x) dx) if possible

a) What is meant by the integrating factor?

b) If the integrating factor looks like e^(ln (x)) how could itbe simplified?

c) What about e^(1/2 ln (x))?

d) What about e^(-ln (x))?

?P(x) dx =

e^(?P(x) dx) =

Step 4: SUBSTITUTE Q(x) from step 2 and e^(?P(x) dx) from step 3into the solution form ye^(?P(x) dx) = ?Q(x)e^(?P(x) dx) + C.

Step 5: INTEGRATE the right side of the result of step 4. Hint:You may need to simplify and/or rewrite the right side before youattempt to integrate.

Step 6: SOLVE for y. To do this divide both sides by theintegrating factor. Remember that all parts of the right sideincluding the C must be divided by the integrating factor.

Solution is _________________________