Find the interval of validity, Mathematics

Assignment Help:

Solve the subsequent IVP and find the interval of validity for the solution.

y' + (4/x) y = x3 y2,       y(2) = - 1,  x > 0

Solution

Thus, the first thing that we require to do is get this into the "proper" form and it means dividing everything via y2.  Doing this provides,

y-2 y' + (4/x) y-1 = x3

The derivative and substitution that we'll require here is as:

n = y-1,                                                n' = -y-2y'

With this type of substitution the differential equation turns into:

- n '+ (4/x ) n = x3

Therefore as noted above it is a linear differential equation which we know how to resolve. We'll do the details on such one and after that for the rest of the illustrations in this section we will leave the details for you to fill in. If you require a refresher on solving linear differential equations so go back to which section for a rapid review.

There is the solution to this differential equation.

n '- (4/x ) n = - x3        ⇒         µ(x) = e-(4/x )dx = e-4In|x| = x-4

∫(x-4n)' dx = ∫-x-1dx       

x4n = - In|x| + c          ⇒         n (x) = cx4 - x4 In x

Remember that we dropped the absolute value bars upon the x in the logarithm due to the assumption that is x >0.

Now we require determining the constant of integration. It can be done in one of two methods. We can change the solution above in a solution in terms of y and after that use the original initial condition or we can change the initial condition into an initial condition in terms of v and then use that. Since we'll need to convert the solution to y's finally anyway and this won't add that a lot work in we'll do this that way.

Therefore, to get the solution in terms of y all we require to do is plug the substitution back in.  Doing it gives:

y-1 = x4 (c - In x)

We can solve for y at this point and after that apply the initial condition or apply the initial condition and after that solve for y. We'll commonly do this with the later approach thus let's apply the initial condition to find:

(-1)-1 = c24 - 24 In 2      ⇒         c= In 2 - 1/16

Plugging in for c and solving for y provides:

1449_Find the interval of validity.png

Remember that we did a little simplification into the solution. It will assists with determining the interval of validity.

Before determining the interval of validity though, we mentioned above which we could convert the original initial condition in an initial condition for n. Let's briefly talk regarding how to do such. To do that all we need to do is plug x = 2 in the substitution and after that use the original initial condition. Doing this provides,

n (2) = y-1(2) = (-1)-1 = -1

Thus in this case we found the same value for v which we had for y. Do not expect that to occur in general if you selected to do the problems in this way.

Okay, let's now determine the interval of validity for the solution. Initially we already identify that x > 0 and it implies that we'll avoid the problems of having logarithms of negative numbers and division through zero at x = 0. Hence, all that we need to worry regarding to then is division by zero in the next term and this will occur where,

1 + 16 In x/2 = 0

⇒ In x/2 = -1/16

⇒ x/2 = e -1/16

⇒ x = 2 e -1/16

≈ 1.8788

The two possible intervals of validity are after that

0 < x < 2 e -1/16

 2 e -1/16 < x < ∞

And as the second one contains the initial condition we identify that the interval of validity is so,

2 e -1/16 < x < ∞

Now there is a graph of the solution.

18_Find the interval of validity1.png


Related Discussions:- Find the interval of validity

.fractions, what is the difference between North America''s part of the tot...

what is the difference between North America''s part of the total population and Africa''s part

Compound interest, Ask question #Minimum 100 words accMick invested $5516 i...

Ask question #Minimum 100 words accMick invested $5516 in an account at 14% compounded quarterly. Calculate the total investment after 1 years.

Linear Systems, Find the solution to the following system of equations usin...

Find the solution to the following system of equations using substitution:

Give the introduction to ratios and proportions, Give the introduction to R...

Give the introduction to Ratios and Proportions? A ratio represents a comparison between two values. A ratio of two numbers can be expressed in three ways: A ratio of "one t

SURFACE AREA AND VOLUMES, Metallic spheres of radii 6 centimetre, 8 centime...

Metallic spheres of radii 6 centimetre, 8 centimetre and 10 centimetres respectively are melted to form a single solid sphere. Find the radius of the resulting sphere.

How to adding polynomials, How to Adding Polynomials? The numerical par...

How to Adding Polynomials? The numerical part of a monomial is called the coefficient. For example, the coefficient of 5x is 5. The coefficient of -7a 2 b 3 is -7. Like

Derive the hicksian demand function using indirect utility , (a) Derive the...

(a) Derive the Marshalian demand functions and the indirect utility function for the following utility function: u(x1, x2, x3) = x1 1/6 x2 1/6 x3 1/6    x1≥ 0, x2≥0,x3≥ 0

Compute the linear convolution, Compute the linear convolution of the discr...

Compute the linear convolution of the discrete-time signal x(n) ={3, 2, 2,1} and the impulse response function of a filter h(n) = {2, 1, 3} using the DFT and the IDFT.

Pie chart, i have this data 48 degree, 72 degree, 43.2degree, 24degree , 40...

i have this data 48 degree, 72 degree, 43.2degree, 24degree , 40.8degree on this make a pie chart

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