Question

9. Question Details ZIDIFEQ9 4.3.009.(38 Find the general solution of the given second-order differential equation. y"+ 36y o y(x) 10. Question Details zomEQ9 4.3.015. Find the general solution of the given higher-order differential equation. yx) - 11.Question Details ZIDTEQ9 4.3.029 Solve the given initial-value problem. y" + 36y-o, y(0)-7, yto)--5 ytx)- 12. Question Details ZIMDifTEQ9 4.4 Solve the given differential equation by undetermined coefficients. y"-6y' + 9y # 6x + 5 y(x)- 13. Question Details ZillDiffE Solve the given differential equation by undetermined coefficients y" + 2y' ty sin(x) +3 cos(2x) y(x)-

Answer 1

"Given differential equation is y"" + 36 y = 0 The auxiliary equation is m^2 + 36 = 0 implies m^2 = -36 implies m = squareroot - 36 = 6 i = 0 + 6 i Therefore the general solution is y (x) = e^0 x (c_1 cos 6 x + c_2 sin 6 x) implies y (x) = c_1 cos 6 x + c_2 sin 6 x) Given y""' - 4 y"" - 5 y' = 0 the auxiliary equation is m^3 - 4 m^2 - 5 m = 0 implies m (m^2 - 4 m - 5) = 0 implies m (m^2 + m - 5 m - 5) = 0 implies m (m (m + 1) - 5 (m + 1)) = 0 implies m (m + 1) (m - 5) = 0 \r\n implies m = 0, - 1, 5 Therefore the general solution is y (x) = c_1 e^0 x + c_2 e^- 1 x + c_3 e^5 x implies y (x)"