Need help on laplace transform and PDE

[reverie414]

Need urgent help on laplace transform and PDE !

I'm stuck with this 2 questions ...

q1) Using laplace transforms, solve: y" + 4y = r(t), where r(t) = {3sint, 0<t<pi, -3sint, t>pi y(0)=0, y'(0)=3.

this is what i get after rewriting for the step function: 3sint [1-u(t-pi)] + (-3sint)u(t-pi) ... I am lost from then on.

q2) Using the method of separation of variables, solve the following partial differential equations:
a)yux(subscript)-xuy(subscript)=0
b)ux(subscript)=yuy(subscript)

i'm really at my wits end ... need to submit shortly afterwards. thanks a lot for the help.

 

[HallsofIvy]

You are at your wit's end but show nothing about what you have tried or what you know? Surely you are not a High School algebra student who has accidently walked into a partial differential equations class!

Don't you have a table of Laplace transforms available? That's the standard method to do such problems. If it is the Laplace transform of a function involving a step function that is bothering you, look at this:
http://www.intmath.com/Laplace/4_lap_laptransunit.php

As far as the partial differential equations are concerned, have you DONE anything at all? "separation of variables" means you try something like u(x,y)= X(x)Y(y). Then u_x= X_x Y and u_y= XY_y. Your first differential equation becomes yX_xY- xXY_y= 0. Dividing through by xyXY makes the equation X_x/(xX)- Y_y/(yY)= 0. Since one part depends only on x and the other only on y, in order to cancel, each must be the same constant:
X_x/(xX)= c and Y_y/yY= c so you have the two ordinary differential equations X_x= cxX and Y_y= cyY. Exactly what possible values c can have will depend on your initial conditions and the actual solution may be a sum of products of X and Y for different values of c.

 

[reverie414]

You are at your wit's end but show nothing about what you have tried or what you know? Surely you are not a High School algebra student who has accidently walked into a partial differential equations class!

i've showed what i attempted as mentioned above. i just really got stuck from there onwards. I've of course tried reading up my textbooks and lecture notes prior to posting. had i fully understand, i wouldn't have posted.

Don't you have a table of Laplace transforms available? That's the standard method to do such problems. If it is the Laplace transform of a function involving a step function that is bothering you, look at this:
http://www.intmath.com/Laplace/4_lap_laptransunit.php

yes i have the laplace transform table and able to do basic inverse laplace transform. what i do not understand is using laplace transform to solve for IVP. an explanation (i can try to work out the answer on my own) about the steps is very much appreciated. thanks for the URL. I am reading it now.

As far as the partial differential equations are concerned, have you DONE anything at all? "separation of variables" means you try something like u(x,y)= X(x)Y(y). Then u_x= X_x Y and u_y= XY_y. Your first differential equation becomes yX_xY- xXY_y= 0. Dividing through by xyXY makes the equation X_x/(xX)- Y_y/(yY)= 0. Since one part depends only on x and the other only on y, in order to cancel, each must be the same constant:
X_x/(xX)= c and Y_y/yY= c so you have the two ordinary differential equations X_x= cxX and Y_y= cyY. Exactly what possible values c can have will depend on your initial conditions and the actual solution may be a sum of products of X and Y for different values of c.

i can't show you what i done for PDE, solely because i do not understand at all.