Try using the Integrating Factor method.
I tried extremely hard on this one. Please help me.
Given: (sint)y' + (cost)y=e^t; y(1)=a, 0<t<pi
a. find the initial value problem and find the critical value a0 exactly. '
so to make this into standard form, I divided by sint.
so: y' + (cost)y/(sint)=e^t/(sint)
μ(t)= e^(ln(sint))
Multiplying both sides by e^(ln(sint)): e^(ln(sint))y' + cost/sint*e^(ln(sint))y=e^t/sint*e^(ln(sint)). Then I got stuck....8(
b. describe the behavior of the solution corresponding to the initial value a0.
c. Draw the direction field of the given differential equation. How do the solutions behave as t goes to 0? Does the behavior depend of the choice of the initial value a? Why or why not? Let a0 be the value for a for which the transition from one type of behavior to another occurs. Estimate the value of a0.
Well, if you follow the program I've laid out for you, you will obtain
To see why this is equivalent to the original DE, just take the derivative on the LHS to get
Hence, if you integrate with respect to time, you get
or, by using the Fundamental Theorem of the Calculus, you get
Can you continue from here? Is everything clear now?