Mathematical Methods for Physicists: A concise introduction - Site Map

THE LAPLACE TRANSFORMATION
9.6 Find
L 1 15
p 2 :
‡ 4p ‡ 13
9.7 Prove that if f 0 …x† is continuous and f 00 …x† is piecewise continuous in every
®nite interval 0 x k and if f …x† and f 0 …x† are of exponential order for
x > k, then
L‰ f F…x†Š ˆ p 2 L‰ f …x†Š pf …0†f 0 …0†:
(Hint: Use (9.19) with f 0 …x† in place of f …x† and f 00 …x† in place of f 0 …x†.)
9.8 Solve the initial problem y 00 …t†‡ 2 y…t† ˆA sin !t; y…0† ˆ1; y 0 …0† ˆ0.
9.9 Solve the initial problem yF…t†y 0 …t† ˆsin t subject to
y…0† ˆ2; y…0† ˆ0; y 00 …0† ˆ1:
9.10 Solve the linear simultaneous di€erential equation with constant coecients
y 00 ‡ 2y x ˆ 0;
x 00 ‡ 2x y ˆ 0;
subject to x…0† ˆ2; y…0† ˆ0, and x 0 …0† ˆy 0 …0† ˆ0, where x and y are the
dependent variables and t is the independent variable.
9.11 Find
Z 1
L cos au du :
9.12. Prove that if L‰ f …x†Š ˆ F…p† then
L‰ f …ax†Š ˆ 1
a F p
:
a
Similarly if L 1 ‰F…p†Š ˆ f …x† then
h
L 1 F p i
ˆ af …ax†:
a
0
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