Four differential equation problems

[seifkhalil]

Hi, I need help in 4 questions. I need their answers if possible.

 

[Simon Bridge]

Hi, I need help in 4 questions. I need their answers if possible.

It helps us to help you if you attempt the problems as well. I'll type them out so you can see how to do it and so as not to inflict a word file on more people than we have to...

1. Solve the following DE:
$$ \frac{d^2y}{dx^2}+3\frac{dy}{dx}+2y=2x+e^{-x} $$

2. Given that $y_1 = e^x$ is a solution of the following homogeneous differential equation :
$$y^{\prime\prime}-3y^\prime+2y=0$$ ...use the method of reduction of order to find a second solution y2 . Hence, find the general solution of the nonhomogeneous differential equation :
$$y^{\prime\prime}-3y^\prime+2y=5e^{3x}$$

3. 3- i) Find the Laplace transform of the following functions:
a) $f(t)= \big ( 1-e^{t/2} \big )^2$
b) $f(t)=\big ( \sin t + \cos t \big )^2$

ii) Find the inverse Laplace transform of the following functions:
a) $$\frac{1}{s^2(s-2)}$$ b) $$\frac{9s+14}{(s-2)(s^2+4)}$$

4. Use Lapalace transform to solve the following IVP:
$$y^{\prime\prime}-6y^\prime+9y=t\; ,\; y(0)=0\; ,\; y^\prime(0)=1$$

Presumably you have notes on how to use the method of reduction of order and the method of laplace transforms? If not - then there are plenty of examples on line.

 

[HallsofIvy]

It helps us to help you if you attempt the problems as well. I'll type them out so you can see how to do it and so as not to inflict a word file on more people than we have to...

Thanks, Simon

1. Solve the following DE:
$$ \frac{d^2y}{dx^2}+3\frac{dy}{dx}+2y=2x+e^{-x} $$

seifkhalil, what do you know about this problem? I presume that you are taking a "Differential Equations" course. Do you recognise this as being a "linear equation with constant coefficients"? Do you understand that the first thing you need to do is find independent solutions to the "associated homogeneous equation" which is
$$ \frac{d^2y}{dx^2}+3\frac{dy}{dx}+2y= 0 $$
Do you know how to find the characteristic equation, $r^2+ 3r+ 2= 0$?
Can you solve that equation? Since one root of the characteristic equation is -1, $e^{-x}$ is a solution to the associated homogenous equation. That means you need to look for a "special" solution to the entire equation of the form "$Ax+ B+ Cxe^{-x}$".

2. Given that $y_1 = e^x$ is a solution of the following homogeneous differential equation :
$$y^{\prime\prime}-3y^\prime+2y=0$$ ...use the method of reduction of order to find a second solution y2 . Hence, find the general solution of the nonhomogeneous differential equation :
$$y^{\prime\prime}-3y^\prime+2y=5e^{3x}$$

Okay, do you know what the "method of reduction or order" is? Since you are told that $e^x$ is as solution, look for another of the form $y(x)= u(x)e^x$. Put that into the given equation and it will reduce to a first order equation for u(x).

3. 3- i) Find the Laplace transform of the following functions:
a) $f(t)= \big ( 1-e^{t/2} \big )^2$
b) $f(t)=\big ( \sin t + \cos t \big )^2$

Do you know what the "Laplace transform" is? The Laplace transform of function f(x) is given by
$$L(f)= \int_0^\infty e^{-st}f(t)dt$$

ii) Find the inverse Laplace transform of the following functions:
a) $$\frac{1}{s^2(s-2)}$$ b) $$\frac{9s+14}{(s-2)(s^2+4)}$$

There is no simple way, such as integration, to find an inverse Laplace transform. You look up the basic ones in a table of transforms. You can typically use "partial fractions" to reduce fractions such as these to the "basic" ones.

4. Use Lapalace transform to solve the following IVP:
$$y^{\prime\prime}-6y^\prime+9y=t\; ,\; y(0)=0\; ,\; y^\prime(0)=1$$

Take the Laplace transform of both sides, which will include "L(y)", the Laplace transform of the uknown function. Algebraically solve the equation for L(y) then use a table of inverse transforms to find y.