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Hi I've just got a quick question. For the following ODE what would be a useful trial function?

y'' + y + y = (sinx)^2

The RHS is sin(x) all squared.

The complimentary solution has trig terms with a root 3 in the arguments so for the trial function I don't need to worry about multiplying by x etc. Usually if I have any linear combination of sine and cosine on the RHS I use y_p(x) = Acos(x) + Bsin(x).

(BTW can I call the trial function the particular integral, seeing as A and B are constants which will be determined anyway.)

This time I have a square of sin(x). So should the trial function be (Acos(x) + Bsin(x))^2? Any help is appreciated.

Edit: Added extra question below.

Does the following question require knowledge of systems of DEs? If it doesn't can someone please help me get started with this one. It seems quite tricky and I haven't yet been able to think of a way to do it.

Q. Consider two tanks A and B, each holding 50 litres of liquid. Pipes connect the tanks with liquid flowing freely between the tanks. Initially, tank A contains 50 litres of salt solution with 25 grams of salt and tank B contains 50 litres of pure water. Pure water is pumped into tank A from an external source at a rate of 3 litres per minute. Salt solution is pumped out of tank B at a rate of 3 litres per minute. Liquid from tank A is pumped into tank B at a rate of 4 litres per minute, while liquid from tank B is pumped into tank A at a rate of 1 litre per minute. Assume that the solutions are well stirred.

(i) Find an expression for the mass of salt in each tank at any given time t.

(ii) What mass of salt will be present in tank A after 50 minutes?

y'' + y + y = (sinx)^2

The RHS is sin(x) all squared.

The complimentary solution has trig terms with a root 3 in the arguments so for the trial function I don't need to worry about multiplying by x etc. Usually if I have any linear combination of sine and cosine on the RHS I use y_p(x) = Acos(x) + Bsin(x).

(BTW can I call the trial function the particular integral, seeing as A and B are constants which will be determined anyway.)

This time I have a square of sin(x). So should the trial function be (Acos(x) + Bsin(x))^2? Any help is appreciated.

Edit: Added extra question below.

Does the following question require knowledge of systems of DEs? If it doesn't can someone please help me get started with this one. It seems quite tricky and I haven't yet been able to think of a way to do it.

Q. Consider two tanks A and B, each holding 50 litres of liquid. Pipes connect the tanks with liquid flowing freely between the tanks. Initially, tank A contains 50 litres of salt solution with 25 grams of salt and tank B contains 50 litres of pure water. Pure water is pumped into tank A from an external source at a rate of 3 litres per minute. Salt solution is pumped out of tank B at a rate of 3 litres per minute. Liquid from tank A is pumped into tank B at a rate of 4 litres per minute, while liquid from tank B is pumped into tank A at a rate of 1 litre per minute. Assume that the solutions are well stirred.

(i) Find an expression for the mass of salt in each tank at any given time t.

(ii) What mass of salt will be present in tank A after 50 minutes?

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