N-th Order Linear Differential Equations 18713. a. Verify that Y1 = ex and Y2 = ex are linearly independent solutions
on (-oo, oo) of the homogeneous linear differential equation(27) y" -y = o.
b. Write the general solution of (27).
c. Find the solution which satisfies the initial conditions y(O) 0,y'(O) = 1.
14. a. Verify that Y1 = 1, Y2 =sin x and y3 =cos x are linearly independent
solutions on ( -oo, oo) of the homogeneous linear differential equa-
tion(28) y'" + y' = 0.
b. Write the general solution of (28).
c. Find the solution which satisfies the initial conditions y(O) 1,y'(O) = 0, y"(O) = -1.
15. a. Verify that y 1 = x and Y2 = x lnx are linearly independent solutions
on (0, oo) of the homogeneous linear differential equation(29) x^2 y" - xy' + y = 0.
b. Write the general solution of (29).
c. Find the solution which satisfies the initial conditions y(l) 2,y'(l) = -1.
16. a. Verify that Yp = 8 is a particular solution on ( -oo, oo) of the non-
homogeneous linear differential equation(30) y" - 4y = 31.
b. Write the associated homogeneous equation.
c. Verify that Y1 = e^2 x and Y2 = e-^2 x are linearly independent solutions
on ( -oo, oo) of the associated homogeneous equation.
d. Write the complementary solution for (30).
e. Write the general solution of (30).
f. Find the solution of (30) which satisfies the initial conditions