Solve the second order differential equations, Mathematics

Assignment Help:

Solve the subsequent IVP

Y'' - 9 y = 0, y(0) = 2, y'(0) = -1

Solution

First, the two functions

 y (t ) = e3t  and  y(t ) = e-3t

That is "nice enough" for us to by the general solution to the differential equation. At this point, please only believe this. You will be capable to verify this for yourself in a couple of sections.

The general solution to our differential equation is now,

y (t ) = c1 e-3t  + c2e3t

Here all we require to do is apply the initial conditions. It means that we require the derivative of the solution.

y' (t ) = -3 e-3t  + 3e3t

Plug in the initial conditions

2 = y (0) = c1 + c2

-1 = y'(0) = -3 c1 +3 c2

This provides us a system of two equations and two unknowns which can be solved.  Doing this yields

c1 = 7/6, c2 = 5/6

The solution to the IVP is so,

 y(t) = (7/6)e-3t + (5/6) e3t

Up to such point we've only looked at a particular differential equation and we found its solution by inspection. For rare little differential equations we can do this. Though, for the huge majority of the second order differential equations out there we will be not capable to do this.

Therefore, we would like a method for arriving at the two solutions we will require so as to form a general solution which will work for any linear, second order differential equation and constant coefficient equation. It is easier than it might initially look.

We will utilize the solutions we found in the first illustration as a guide. Each of the solutions in this illustration were in the form

y (t ) = ert

Remember, that we didn't include a constant in front of it as we can literally comprise any constant which we want and still get a solution. The significant idea here is to find the exponential function. One time we have which we can add on constants to our hearts content.

Thus, let's suppose that all solutions to,

ay′′ + by′ + cy = 0   ...................(4)

It will be of the form as

y (t ) = e rt        ........................(5)

 

To notice if we are correct all we require to do is plug this in the differential equation and notice what occurs.  Thus, let's get several derivatives and after that plug in.

y′ (t )= rert

 y′′ (t ) = r2ert

 a (r2ert )+ b (rert )+ c (ert ) = 0

ert (ar2 + br + c )= 0

Therefore, if (5) is to be a solution to (4) then the subsequent must be true

ert (ar2 + br + c ) = 0

It can be decreased further by noting as exponentials are never zero. Thus, (5) will be a solution to (4) given r is a solution to

 ar2 + br + c = 0    .......... (6)

 This equation is usually termed as the characteristic equation for (4).

Okay, then how do we use this to get solutions to a linear, constant coefficient, second order differential equation? First write down the feature equation, (6), for the differential equation, (4). It will be a quadratic equation and thus we must expect two roots, r1 and r2. One time we have these two roots we have two solutions to the differential equation.

1538_Solve the SECOND ORDER DIFFERENTIAL EQUATIONS.png...............(7)


Related Discussions:- Solve the second order differential equations

School mathematics, I am interested in school mathematics online assignment...

I am interested in school mathematics online assignments , homework help, projects etc. I have good knowledge of mathematics and experience of 15+ years teaching mathematics in cen

Example of complex roots, Solve the subsequent IVP. y'' - 4y' + 9y = 0, ...

Solve the subsequent IVP. y'' - 4y' + 9y = 0, y(0) = 0, y'(0) = -8 Solution The characteristic equation for such differential equation is. As:  r 2 - 4r + 9 = 0

Compute the center of mass of the solid, 1) Compute the center of mass of t...

1) Compute the center of mass of the solid of unit density 1 bounded (in spherical coordinates) by p=1 and by φ is greater than or equal 0 and less than or equal pi/4

Regression, Regression line drawn as y=c+1075x, when x was 2, and y was 239...

Regression line drawn as y=c+1075x, when x was 2, and y was 239, given that y intercept was 11. Caculate the residual

How many 6-inch tiles are required to tile the floor, Mark intends to tile ...

Mark intends to tile a kitchen floor, which is 9 by 11 ft. How many 6-inch tiles are required to tile the floor? a. 60 b. 99 c. 396 c. Since the tiles are calculated in

Applications of percentage, rajan bought an armchair for rs.2200 and sold i...

rajan bought an armchair for rs.2200 and sold it for rs.2420.find his profit per cent.

#title LOGIC, HOW MANY ZERO ARE THERE AT THE END OF 200

HOW MANY ZERO ARE THERE AT THE END OF 200

Area and perimeter, if perimeter is 300m length is 100m.find the breadth

if perimeter is 300m length is 100m.find the breadth

Example of spiral development of the mathematics curriculum?, E1) Can you g...

E1) Can you give some more examples of the spiral development of the mathematics curriculum? E2) A Class 3 child was asked to add 1/4 + 1/5. She wrote 2/9. Why do you feel this

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