MATH 202 Final, Spring 2004

LIR TARY oF PreRegr MATH 202: FINAL EXAM, JUNE 2npD 2004 PROFESSOR: IVA STAVROV Problem #1. Find the general solution of the given system af 2°) - Pata 2) Problem #2. Use the Laplace transform to solve the given integrodifferential equation. ove) + f vluidu = 6(t 2) y= Problem #3. Use the Laplace transform to solve the given ]VP. y+ 4y= ft). yO) =y(0) =1 Here the function f is given by et ift<), fos 0 if1<i<2, ” 1 yi tym cos? 0)== yO) = 5 & 1 (x)= — We) = OR Problem #5. Use a power series centered at 2 = 0 to solve the following differ- ential equation. You should find all the coefficients in the general solution unti] the term involving x* (inclusive). You do not have to find the general formula for the coefficients. yl” wy = 2. Problem #6. Use the substitution y = r~!z(x) to transform the differential equation 4x7y" + 12ay’ + (42? + 3)y = 0 into a Bessel’s differential equation. Then express the general solution of the above equation in terms of Bessel functions. Problem #7. It is known that there exists m such that u(r, y) = (x + y)” is the integrating factor for (40 + y + 8)dx + (3x + 3)dy = 0. Find n and use it to solve the given equation. nN Problem #8. (1) Find at least two solutions of the IVP: { y = 3y y(0) = 0. (2) State the existence and uniqueness theorem related to the 1VP given in this problem. 3) Explain why the fact that our ]VP has more than one solution does not } I contradict the theorem.