posted by .

Solve the IVP

{y'=-(sinx)y + xexp(cosx)
y(0) = 1

Please show me step by step. Do I need to find out if this is exact? Do I use an integrating factor?

  • Calculus -

    Solve the homogeneous equation first:

    y_h' = - sin(x) y_h ---->

    y_h(x) = K exp[cos(x)]

    Then, to find the solution, you use the variation of the constant method, i.e. take the solution of the homogeneous solution y_h and there you replace K by an unknown function K(x):

    y(x) = K(x) exp[cos(x)]

    If you substitute this in the differential equation then what happens is that only the term proportional to K' survives. Due to the product rule the terms proportional to K are what you would get if K were a constant, but these terms will satisfy the homogeneous equation, so they will sum to zero.

    So, what you get is:

    K' Exp[cos(x)] = x Exp[cos(x)] ----->

    K = 1/2 x^2 + c ----->

    y(x) = (1/2 x^2 + c)Exp[cos(x)]

Respond to this Question

First Name
School Subject
Your Answer

Similar Questions

  1. Trig........

    I need to prove that the following is true. Thanks (cosx / 1-sinx ) = ( 1+sinx / cosx ) I recall this question causing all kinds of problems when I was still teaching. it requires a little "trick" L.S. =cosx/(1-sinx) multiply top and …
  2. Pre-Calculus

    How do you verify: 1.) sinxcosx+sinx^3secx=tanx 2.) (secx+tanx)/(secx-tan)=(secx+tanx)^2 I tried starting from the left on both problems, but am stuck. like I said before, change everything into sines and cosines. for the first one: …
  3. Pre-calc

    prove the identity: (cosx)(tanx + sinx cotx)=sinx+cos(squared)x i need steps to show how i got the answer generally, it is a good idea to change all trig ratios to sines and cosines, and start with the more complicated-looking side. …
  4. math

    the problem is 2cos^2x + sinx-1=0 the directions are to "use an identity to solve each equation on the interval [0,2pi). This is what i've done so far: 2cos^2x+sinx-1=0 2cos^2x-1+sinx=0 cos2x + sinx =0 1 - 2sin^2x + sinx = 0 -2sin^2x+sinx-1=0 …
  5. grouping

    i am trying to solve for A and B....this question comes from a Differential equations...but this step goes back to basics... Asinx-Bcosx+6Acosx+6Bsinx=3-cosx when you group all the sinx and cosx you get (A+6B)sinx+(6A-B)cosx=3-cosx …
  6. math

    solve each equation for 0=/<x=/<2pi sin^2x + 5sinx + 6 = 0?
  7. maths - trigonometry

    I've asked about this same question before, and someone gave me the way to finish, which I understand to some extent. I need help figuring out what they did in the second step though. How they got to the third step from the second. …
  8. Calculus

    Determine the limit of the trigonometric function: Limit as x approaches pi/4 of (1-tanx)/(sinx-cosx) I need step by step help using the special trignometric limits theorem.
  9. trigonometry

    can i use factoring to simplify this trig identity?
  10. Algebra 1--Step-by-Step

    Can someone show me how to solve these step-by-step?

More Similar Questions