Differential equation to determine initial value problem, Mathematics

Assignment Help:

Solve the subsequent IVP.

cos(x) y' + sin(x) y = 2 cos3(x) sin(x) - 1

y(p/4) = 3√2, 0 < x< p/2

Solution:

Rewrite the differential equation to determine the coefficient of the derivative an individual.

y' + (sin(x)/cos(x))y = 2cos2 (x) sin(x) - 1/cos(x)

y' + tan(x)y = 2cos2 (x) sin(x) - sec(x)

Now determine the integrating factor:

1689_Differential equation to determine initial value problem.png

Can you do the integral? If not rewrite tangent back in sines and cosines and after that use a easy substitution. Remember that we could drop the absolute value bars upon the secant due to the limits on x.  Actually, this is the purpose for the limits on x.

Also remember that we made use of the subsequent fact.

eInf(x) = f(x)    .........................(11)

It is a significant fact that you must always keep in mind for these problems. We will want to make simpler the integrating factor as much as probable in each case and this fact will assist with which simplification.

Currently back to the illustration. Multiply the integrating factor by the differential equation and confirm the left side is a product rule. Notice also that we multiply the integrating factor by the rewritten differential equation and NOT the original differential equation. Ensure that you do that. If you multiply the integrating factor via the original differential equation you will find out the wrong solution!

sec(x) y' + sec(x) tan (x)y = 2sec(x) cos2(x) sin(x) - sec2(x)

(sec(x) y)' = 2cos(x) sin(x) -sec2(x)

Integrate both sides.

∫(sec(x) y)' dx = ∫(2cos(x) sin(x) -sec2(x)) dx

sec(x) y(x) = ∫ sin(2x) - sec2(x) dx

sec(x) y(x) = - ½  cos(2x) - tan(x) + c

See there the use of the trig formula sin (2q) = 2 sin q cosq resolve for the solution.

y(x) = - ½ cos(x) cos(2x) - cos(x) tan(x) + c cos(x)

= - ½ cos(x) cos(2x) - sin(x) + c cos(x)

At last, apply the initial condition to determine the value of c.

 

1146_Differential equation to determine initial value problem1.png

The solution is afterward as:

y(x) =  - ½ cos(x) cos(2x) - sin(x) + 7 cos(x)

A plot of the solution is here given below:

2202_Differential equation to determine initial value problem2.png


Related Discussions:- Differential equation to determine initial value problem

Linear Programming, A garden shop wishes to prepare a supply of special fer...

A garden shop wishes to prepare a supply of special fertilizer at a minimal cost by mixing two fertilizers, A and B. The mixture is to contain at least 45 units of phosphate at lea

Exponent, base also called what

base also called what

Calculate one-sided limits, Calculate the value of the following limits. ...

Calculate the value of the following limits. Solution From the graph of this function illustrated below, We can illustrate that both of the one-sided limits suffer

Tangent, construction of tangent when center not known

construction of tangent when center not known

Explain equation, Equation s(in Tth second)=u+at-a/2 seems to be dimensiona...

Equation s(in Tth second)=u+at-a/2 seems to be dimensionally incorrect.why?

Area related to circle, If ABCD isaa square of side 6 cm find area of shad...

If ABCD isaa square of side 6 cm find area of shaded region

Proof of sum-difference of two functions, Proof of Sum/Difference of Two Fu...

Proof of Sum/Difference of Two Functions : (f(x) + g(x))′  = f ′(x) +  g ′(x)  It is easy adequate to prove by using the definition of the derivative.  We will start wi

Probability distributions, Probability Distributions Since the value of...

Probability Distributions Since the value of a random variable cannot be predicted accurately, by convention, probabilities are assigned to all the likely values that the varia

Testing the hypothesis equality of two variances, Testing the hypothesis eq...

Testing the hypothesis equality of two variances The test for equality of two population variances is based upon the variances in two independently chosen random samples drawn

About matrix?, Explain sparse matrix and Dense matrix?

Explain sparse matrix and Dense matrix?

Write Your Message!

Captcha
Free Assignment Quote

Assured A++ Grade

Get guaranteed satisfaction & time on delivery in every assignment order you paid with us! We ensure premium quality solution document along with free turntin report!

All rights reserved! Copyrights ©2019-2020 ExpertsMind IT Educational Pvt Ltd