1 Obtain the General Solution of the differential equations:
(i)
(ii)
2 (a) Find the general solution of the differential equation
(b) Find the solution of
given that at x=0,y=2 and
3 (a) Find the following two integrals:
and
Find y in terms of x, givene that
and that y =1 when x = 0.
(b) Solve the differential equation
given that y=1 and when x=0.
Divide through by the coefficient of to get
Apply the technique of the integrating factor here. Can you continue?
Do you mean or ??(ii)
Divide through by the coefficient of to get2 (a) Find the general solution of the differential equation
Now apply the technique of the integrating factor. Can you continue from here?
The characteristic equation is which means that (Verify)(b) Find the solution of
given that at x=0,y=2 and
Thus, the homogeneous solution is
Now apply the method of undetermined coefficients to determine the particular solution. Once you find the general solution (homogeneous + particular), apply the initial conditions. Can you take it from here?
They're the same thing....3 (a) Find the following two integrals:
and
Apply the substitution ...
You should do this with no problem once you have figured out to integrate . Again, you're using the technique of the integrating factor to solve this.Find y in terms of x, givene that
and that y =1 when x = 0.
The corresponding characteristic equation is which yields (Verify)(b) Solve the differential equation
given that y=1 and when x=0.
The general solution would be
Can you take it from here and apply the initial conditions?
Does this make sense?
2. a) This is a first order linear ODE.
So we use the Integrating Factor method.
Divide everything through by x first, to get
.
Now we find the Integrating Factor and multiply both sides of the equation by it.
It's .
So the DE becomes
.
The left hand side is the product rule expansion of
So
.
Once again, the RHS can be evaluated using Integration by Parts.
Thanks for the replies. Regarding the - and + it was - if I remember correctly, and those weren't supposed to be the same thing >.< There was some minor difference.
Alas, I woke up late and couldn't get these done so... i'm as good as roasted, but that's completely my fault for not switching ISPs and for being a blockhead >.>
Thanks for all the inputs, and Pivot thanks for explaining Integrating Factor, I was lost with those for the time being. I may get yelled at but at least I understood 'em xD