Questions on basic differential equations

[danield]

Well this is my first diff eqs homework and I am totally lost, i have no idea what to do here are the questions that i have...

Homework Statement

1)An object released from a height h meters above the gorund with a veritcal velocity of Vo m/s htis the ground after To seconds. Neglecting fricitonal forces set up and solve the inital value problem governing the motion and use your solution to show that...
Vo=(2h-gto^2)/(2To)

2) Determine a solution to the differential equation
(1-x^2)y'' - xy' +4y = 0
of the form Y(x) = a0 + a1x + a2x^2 satisfying the normalization condition y(1) = 1

3) Determine the differential equaiton giving the slope of the tangent line at the point (x,y) for the given curve
x^2 + y^2 = 2cx

The Attempt at a Solution

1)on that one i got till the point where i have
1/2gt^2 + C1t+ c2 = y(t)


2) On this one i have no idea how to start maybe help on where to start will be enough...
3) on this one i also have no idea where to start...

im still working on other ones ill see if I am able to work them out


any help is appreciated
thanks

 

[HallsofIvy]

Well this is my first diff eqs homework and I am totally lost, i have no idea what to do here are the questions that i have...

Homework Statement

1)An object released from a height h meters above the gorund with a veritcal velocity of Vo m/s htis the ground after To seconds. Neglecting fricitonal forces set up and solve the inital value problem governing the motion and use your solution to show that...
Vo=(2h-gto^2)/(2To)

2) Determine a solution to the differential equation
(1-x^2)y'' - xy' +4y = 0
of the form Y(x) = a0 + a1x + a2x^2 satisfying the normalization condition y(1) = 1

3) Determine the differential equaiton giving the slope of the tangent line at the point (x,y) for the given curve
x^2 + y^2 = 2cx

The Attempt at a Solution

1)on that one i got till the point where i have
1/2gt^2 + C1t+ c2 = y(t)

Okay, you know that y(0)= (1/2)g(0^2)+ C1(0)+ C2= h (An object released from a height h meters above the ground) and that y'(0)= g(0)+ C1= V0 (with a vertical velocity of Vo) so you can determine C1 and C2 from those.

2) On this one i have no idea how to start maybe help on where to start will be enough...

Do it! You are asked for a solution of the form $y= a_0+ a_1x+ a_2x^2$ so $y'= a_1+ 2a_2x$ and $y"= 2a_2$. Put those into the differential equation and solve for $a_0$, $a_1$, and $a_2$ by, for example, choosing three values for x to get three equations.

3) on this one i also have no idea where to start...

You have a "family" of curves given by $x^2+ y^2= 2cx$ and you want an equation involving y' and/or y" but not c. Now, you could just differentiate with respect to x twice and that would eliminate c: $2x+ 2y y'= 2c$ and then $2+ 2(y')^2+ 2y y"= 0$. But since there is only one "c" you should be able to do this with just one differentiation. Differentiating once gives $2x+ 2y y'= 2c$ so $c= x+ y y'$. Replace c in $x^2+ y^2= 2cx$ with that!

im still working on other ones ill see if I am able to work them out


any help is appreciated
thanks