MATH215 Lecture Notes - Lecture 1: Integrating Factor

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violethare11

13 Mar 2017

School

University of Alberta

Department

Mathematics

Course

MATH215

Professor

Alexander Litvak

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Document Summary

Di erential equations: a rst order di erential equation has the form g(x) h(y) it is called a separable equation and can be solved by the method of separation of variables: Z h(y) dy = z g(x) dx: a rst order linear di erential equation is of the form. To solve the equation, we multiply both sides of the equation by the integrating factor dy dx. The equation becomes d dx(cid:0)i(x)y(cid:1) = i(x)q(x) or. I(x)y = z i(x)q(x) dx: a second order homogeneous di erential equation with constant coe cients has the form ay + by + cy = 0 (a 6= 0). A particular solution can be found by using the method of variation of parameters: the method of variation of parameters always works. Let y1 and y2 be two linearly independent solutions of the corresponding homogeneous equation ay + by + cy = 0. If u1 and u2 satisfy the equations u .

Related Questions

Please help me with this problem. I already done the first part.but the second one is neither separable nor linear. I have troublewith this.

Thx!

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brownhamster745

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 _________________________

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MATH215 Lecture Notes - Lecture 1: Integrating Factor

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