Chapter 10.4, Problem 1TFR

For Questions (a)-(d), decide if the given statement is true or false, and give a brief justification for your answer. If true, you can quote a relevant definition or theorem in fact from the text. If false, provide an example, illustration, or brief explanation of why the statement is false.

(a) For every function $f$ with a continuous derivative on $\left[0,\infty \right)$, the Laplace transform of the derivative is given by $L\left[f^{\prime }\right]=sL\left[f\right]-f\left(0\right)$.

(b) In solving an initial value problem using the Laplace transform method, the general solution of the differential equation is not explicitly found. Rather, we impose the initial conditions immediately in the procedure.

(c) The Laplace transform method for solving an initial value problem can also be used to find the general solution of the differential equation.

(d) The initial conditions of an initial-value problem do not affect the expression $Y\left(s\right)$ for the Laplace transform of the solution $y\left(t\right)$.