[solved] solve them1. Solve the initial value problem y 00 − 6y 0 + 9y = 0 with y(0) = 0 y 0 (0) = 2. 2. Find the general solution of y 00 + 6y 0 + 13y = 0. 3. The techniques and theory we have developed in Chapter 3 are naturally extended to higher order LHCC differential equations. Extend the methodology of Chapter 3 to find the general solution of 2y 000 − 4y 00 − 2y 0 + 4y = 0. 4. y1 = t 2 and y2 = 1 t are solutions of the differential equation t 2 y 00 − 2y = 0 for t > 0. Use the Wronskian to determine if these two solutions can be used as a fundamental set of solutions for this differential equation. 5. Create your own second order LHCC differential equation such that as t → ∞ the solutions grow without bound for some choices of initial conditions but the solutions approach zero for other choices of initial conditions. Please clearly state your differential equation AND its general solution. 6. Consider a general second order LHCC differential equation ay00 + by0 + cy = 0. a) Show that if a > 0 b > 0 and c > 0 then limt→∞ y = 0 for all solutions y of this differential equation. b) If a > 0 b > 0 but c = 0 find the general solution of the resulting differential equation. In addition compute limt→∞ y for solutions of the differential equation. 7. Consider the non-homogeneous differential equation y 00 − 3y 0 − 4y = g(t). For each g(t) given below state the appropriate choice of form for yp that you would use to implement the method of undetermined coefficients. You do NOT need to solve for the coefficients in your yp; you do NOT need to state the solution of the original differential equation. a) g(t) = −e −t + e −t cos(t) b) g(t) = 2te4t + e 3t 8. Use variation of parameters to find the general solution of the differential equation y 00+3y 0−10y = e kt where k is an arbitrary constant.

[solved] solve them1. Solve the initial value problem y 00 − 6y 0 + 9y = 0 with y(0) = 0 y 0 (0) = 2. 2. Find the general solution of y 00 + 6y 0 + 13y = 0. 3. The techniques and theory we have developed in Chapter 3 are naturally extended to higher order LHCC differential equations. Extend the methodology of Chapter 3 to find the general solution of 2y 000 − 4y 00 − 2y 0 + 4y = 0. 4. y1 = t 2 and y2 = 1 t are solutions of the differential equation t 2 y 00 − 2y = 0 for t > 0. Use the Wronskian to determine if these two solutions can be used as a fundamental set of solutions for this differential equation. 5. Create your own second order LHCC differential equation such that as t → ∞ the solutions grow without bound for some choices of initial conditions but the solutions approach zero for other choices of initial conditions. Please clearly state your differential equation AND its general solution. 6. Consider a general second order LHCC differential equation ay00 + by0 + cy = 0. a) Show that if a > 0 b > 0 and c > 0 then limt→∞ y = 0 for all solutions y of this differential equation. b) If a > 0 b > 0 but c = 0 find the general solution of the resulting differential equation. In addition compute limt→∞ y for solutions of the differential equation. 7. Consider the non-homogeneous differential equation y 00 − 3y 0 − 4y = g(t). For each g(t) given below state the appropriate choice of form for yp that you would use to implement the method of undetermined coefficients. You do NOT need to solve for the coefficients in your yp; you do NOT need to state the solution of the original differential equation. a) g(t) = −e −t + e −t cos(t) b) g(t) = 2te4t + e 3t 8. Use variation of parameters to find the general solution of the differential equation y 00+3y 0−10y = e kt where k is an arbitrary constant.