Follow the six steps below to solve the following: xy\'+6xy=xe^5x Step 1: REWRIT

Question

Follow the six steps below to solve the following: xy'+6xy=xe^5x Step 1: REWRITE into standard form (y'+yP(x)+Q(x)) a) How does the given equation differ from standard form? b) What can be done to put the given equation into standard form? Step 2: IDENTIFY P(x) and Q(x). a) Where does the standard form indicate P(x) and Q(x) are located? P(x) = Q(x) = Step 3: DETERMINE the integrating factor (e^?P(x)dx). Plug P(x) as identified in step 2 into the integrating factor form e^?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 it be 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 3 into 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 you attempt to integrate. Step 6: SOLVE for y. To do this divide both sides by the integrating factor. Remember that all parts of the right side including the C must be divided by the integrating factor.

Explanation / Answer

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

y' + 6y = e^5x

now

integrating factor

e^ integral of 6dx = e^6x

now

solution is

y e^6x = integral of e^6x e^5x dx

y e^6x = integral of e^11x dx

y e^6x = e^11x/11 + C

y = e^5x/11 + C e^-6x

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