Laplace Theory Examples Harmonic oscillator s-differentiation Rule First shifting rule Trigonometric formulas Exponentials Hyperbolic functions s-differentiation Rule First Shifting Rule I and II Damped oscillator Second Shifting Rule I and II
1 Example (Harmonic oscillator) Solve the initial value problem x + x 0, x(0) 0, x (0) 1 by Laplace s method. The solution is x(t) sin t. The details: L(x ) + L(x) L(0) Apply L across the equation. sl(x ) x (0) + L(x) 0 Parts rule. s[sl(x) x(0)] x (0) + L(x) 0 Parts rule again. (s + 1)L(x) 1 Use x(0) 0, x (0) 1. L(x) 1 s + 1 Isolate L(x) left. L(sin t) Backward Laplace table. x(t) sin t Lerch s cancellation law.
Example (s-differentiation Rule) Show the steps for L(t e 5t ) (s 5) 3. ( L(t e 5t ) d ) ( d ) L(e 5t ) Use s-differentiation. ds ds ( 1) d ( ) d 1 ds ds s 5 d ( ) 1 Calculus power rule. ds (s 5) Identity verified. (s 5) 3
3 Example (First shifting rule) Show the steps for L(t e 3t ) (s + 3) 3. L(t e 3t ) L(t ) s s ( 3) ( ) s +1 s s ( 3) First shifting rule. Identity verified. (s + 3) 3
4 Example (Trigonometric formulas) Show the steps used to obtain these Laplace identities: (a) L(t cos at) s a (c) L(t cos at) (s3 3sa ) (s + a ) (s + a ) 3 sa (b) L(t sin at) (d) L(t sin at) 6s a a 3 (s + a ) (s + a ) 3 The details for (b) and (d) are left as exercises. The details for (a): L(t cos at) (d/ds)l(cos at) d ( ) s ds s + a The details for (c): Use s-differentiation. s a (s + a ) Calculus quotient rule. L(t cos at) (d/ds)l(( t) cos at) d ( ) s a ds (s + a ) Use s-differentiation. Result of (a). s3 6sa ) (s + a ) 3 Calculus quotient rule.
5 Example (Exponentials) Show the steps used to obtain these Laplace identities: s a (a) L(e at cos bt) (c) L(te at cos bt) (s a) b (s a) + b ((s a) + b ) b b(s a) (b) L(e at sin bt) (d) L(te at sin bt) (s a) + b ((s a) + b ) Left as exercises are (a), (b) and (d). Details for (c): L(te at cos bt) L(t cos bt) s s a ( d ) ds s s a L(cos bt) ( d ( )) s ds s + b ( ) s b (s + b ) (s a) b ((s a) + b ) s s a s s a First shifting rule. Use s-differentiation. Calculus quotient rule. Verified (c).
6 Example (Hyperbolic functions) Establish these Laplace transform facts about cosh u (e u + e u )/ and sinh u (e u e u )/. s (a) L(cosh at) (c) L(t cosh at) s + a s a (s a ) a as (b) L(sinh at) (d) L(t sinh at) s a (s a ) Left as exercises are (b) and (c). The details for (a): L(cosh at) 1 (L(eat ) + L(e at )) 1 ( 1 s a + 1 ) s + a s s a The details for (d): L(t sinh at) d ds ( a s a ) Definition and linearity. Identity (a) verified. Apply the s-differentiation rule. a(s) (s a ) Calculus power rule; (d) verified.
7 Example (s-differentiation Rule) Solve L(f(t)) The solution is f(t) t sin t. The details: s L(f(t)) (s + 1) d ( ) 1 ds s + 1 s for f(t). (s + 1) Calculus power rule (u n ) nu n 1 u. d ds (L(sin t)) Backward Laplace table. L(t sin t) Apply the s-differentiation rule. f(t) t sin t Lerch s cancellation law.
s + 7 8 Example (First Shifting Rule I) Solve L(f(t) for f(t). s + 4s + 8 The answer is f(t) e t (cos t + 5 sin t). The details: s + 7 L(f(t)) Complete the square. (s + ) + 4 S + 5 Replace s + by S. S + 4 S S + 4 + 5 Split into table entries. S + 4 s s + 4 + 5 s + 4 Shifting rule preparation. s Ss+ L ( cos t + 5 sin t) s Ss+ L(e t (cos t + 5 sin t)) First shifting rule. f(t) e t (cos t + 5 sin t) Lerch s cancellation law.
s + 9 Example (First Shifting Rule II) Solve L(f(t)) for f(t). + s + The answer is f(t) e t cos t + e t sin t. The details: s + L(f(t)) Signal for this method: the denominator has complex roots. s + s + s + Complete the square, denominator. (s + 1) + 1 S + 1 Substitute S for s + 1. S + 1 S S + 1 + 1 Split into Laplace table entries. S + 1 L(cos t) + L(sin t) s Ss+1 L(e t cos t) + L(e t sin t) First shift rule. f(t) e t cos t + e t sin t Invoke Lerch s cancellation law.
10 Example (Damped oscillator) Solve by Laplace s method the initial value problem x + x + x 0, x(0) 1, x (0) 1. The solution is x(t) e t cos t. The details: L(x ) + L(x ) + L(x) L(0) Apply L across the equation. sl(x ) x (0) + L(x ) + L(x) 0 The t-derivative rule on x. s[sl(x) x(0)] x (0) The t-derivative rule on x. +[L(x) x(0)] + L(x) 0 (s + s + )L(x) 1 + s Use x(0) 1, x (0) 1. s + 1 L(x) s + s + Divide to isolate L(x). s + 1 Complete the square in the denominator. (s + 1) + 1 L(cos t) s s+1 L(e t cos t) First shifting rule. x(t) e t cos t Lerch s cancellation law.
11 Example (Second Shifting Rule I) Show the steps for L(sin t H(t π)) e πs s + 1. The second shifting rule is applied as follows. LHS L(sin t H(t π)) Left side of the identity. L(g(t)H(t a)) Choose g(t) sin t, a π. e as L(g(t + a) Second form, second shifting theorem. e πs L(sin(t + π)) Substitute a π. e πs L( sin t) Sum rule sin(a + b) sin a cos b + sin b cos a plus sin π 0, cos π 1. 1 e πs s + 1 RHS Identity verified.
1 Example (Second Shifting Rule II) Solve L(f(t)) e 3s s + 1 s + s + for f(t). The answer is f(t) e 3 t cos(t 3)H(t 3). The details: s + 1 L(f(t)) e 3s (s + 1) + 1 Complete the square. S e 3s S + 1 Replace s + 1 by S. e 3S+3 (L(cos t)) s Ss+1 e 3 ( e 3s L(cos t) ) s Ss+1 Regroup factor e 3S. e 3 (L(cos(t 3)H(t 3))) s Ss+1 Second shifting rule. e 3 L(e t cos(t 3)H(t 3)) First shifting rule. f(t) e 3 t cos(t 3)H(t 3) Lerch s cancellation law.