a gradient of a curve at point (x,y) is given by sin x-y the point (pai, 1) lies on the curve. determine the equation of the curve in the form of y = f(x)
in Calculus Answers by

Your answer

Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
To avoid this verification in future, please log in or register.

1 Answer

dy/dx=sin(x)-y, dy/dx+y=sin(x).

Multiply through by eˣ:

eˣdy/dx+yeˣ=eˣsin(x)

d(yeˣ)/dx=eˣsin(x)

Integrate:

yeˣ=∫eˣsinxdx+k where k is a constant.

Let u=eˣ, dv=sinxdx then du=eˣ, v=-cosx. Let J=∫eˣsinxdx=-eˣcosx+∫eˣcosxdx.

Let dv=cosxdx, then v=sinx, and ∫eˣcosxdx=eˣsinx-∫eˣsinxdx=eˣsinx-J.

So J=-eˣcosx+eˣsinx-J, 2J=eˣ(sinx-cosx), J=½eˣ(sinx-cosx)=yeˣ-k.

But when x=π, y=1, so k=e^π-½e^π(-(-1))=½e^π.

yeˣ=½eˣ(sinx-cosx)+½e^π

y=f(x)=½(sinx-cosx)+½e^(π-x).

CHECK:

dy/dx=½(cosx+sinx)-½e^(π-x)

y=½(sinx-cosx)+½e^(π-x)

Add these equations together:

dy/dx+y=sinx so this confirms the solution.

And when x=π, y=½+½=1.

by Top Rated User (610k points)

Related questions

Welcome to MathHomeworkAnswers.org, where students, teachers and math enthusiasts can ask and answer any math question. Get help and answers to any math problem including algebra, trigonometry, geometry, calculus, trigonometry, fractions, solving expression, simplifying expressions and more. Get answers to math questions. Help is always 100% free!
82,170 questions
86,660 answers
2,246 comments
76,278 users