Page 264 - Schaum's Outline of Differential Equations

CHAP. 24] SOLUTIONS OF LINEAR DIFFERENTIAL EQUATIONS 247

24.14. Solve = 0.05N; N(0) = 20,000.

This is a differential equation for the unknown function N(t) in the independent variable t. We set
N(s) = !£{N(t)}. Taking Laplace transforms of the given differential equation and using (24.4) with N replacing y,
we have

or, upon solving for N(s),

Then from Appendix A, entry 7 with a = 0.05 and t replacing x, we obtain

Compare with (2) of Problem 7.1.

24.15. Solve + 507 = 5; 7(0) = 0.

This is a differential equation for the unknown function I(t) in the independent variable t. We set I(s) = ££{/(£)}.
Taking Laplace transforms of the given differential equation and using Eq. (24.4) with 7 replacing y, we have

or, upon solving for I(s),

Then using the method of partial fractions and Appendix A, with t replacing x, we obtain

Compare with (1) of Problem 7.19.

24.16. Solve x + 16x = 2sm4t;x(0) = -±,x(0) = 0.

This is a differential equation for the unknown function x(t) in the independent variable t. We set
X(s) = ££{XO)- Taking Laplace transforms of the given differential equation and using Eq. (24.5) with x replacing
y, we have