Part I: Separable Differential Equations 1. What is meant by separable? What sho

Question

Part I: Separable Differential Equations 1. What is meant by separable? What should you look for in a differential equation to decide if it is separable or not? 2. Is y^'-6x=4 separable? If so, how can it be separated? Hint: Remember that y^'=dy/dx. 3. Integrate both the left and right sides of the result of question 2. 5. Can the result of question 3 be easily solved for y? If so, solve for y. If not, how should we write the final answer? Under what circumstances would the integration result in an equation that should not be solved for y? Solution is _____________________ 6. Solve the following separable differential equation by completing the three step process. ?2y^3 e?^x dy-ye^5x dx=0 SEPARATE

Explanation / Answer

separable means u can seperate x and dx on one side and y and dy on other side and just integrate

y^'-6x=4

Yeah it is separable

-6x ln y = 4

=> lny = -2/ (3x)

so separable

now integratke

=> 1/y y ' = -2/3 lnx

=>y ' = (-2y/3) lnx + c

y ' +yP(x) = Q(x))

this is another type

here we find integrating factor = e^ integral ( p(x))

and solution will be y IF = integral ( IF q(x))

?xy?^'+6xy=xe^5x

y + x y ' + 6xy = xe^5x

=>xy ' + y(6x+1) = x e^5x

=> y ' + y (6 + 1/x) = e^5x

now IF = e^ (Integral 6 + 1/x = e^(6x + lnx) = e^6x .x

so solution is

y(xe^(6x)) = integral ((xe^(6x)) *e^5x )

=>y(xe^(6x)) = integral ( x e^11x)

=>y(xe^(6x)) = x integral(e^11x) - integral ( 1 * integral(e^11x) )

=>y(xe^(6x)) = = xe^11x /11 - e^11x / 121

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