ENGR 2022 M3 Laplace Transform Complete study Questions and Answers

Laplace Transform
Applied Mathematics - III




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, LAPLACE TRANSFORM
Type I: Basic
1. Find the Laplace Transform of ‫ݐ͵݊݅ݏݐ݊݅ݏݐʹ݊݅ݏ‬
[N13/ChemBiot/5M]
Solution:
‫ܮ‬ሼ‫ݐ͵݊݅ݏݐ݊݅ݏݐʹ݊݅ݏ‬ሽ
ͳ
ൌ ‫ܮ‬ሼʹ‫ ݐ͵݊݅ݏݐ݊݅ݏݐʹ݊݅ݏ‬ሽ
ʹ
ͳ
ൌ ‫ ܮ‬ሼሺܿ‫ ݐݏ݋‬െ ܿ‫ݐ͵ݏ݋‬ሻ‫ ݐ͵݊݅ݏ‬ሽ
ʹ
ͳ
ൌ ‫ܮ‬ሼܿ‫ ݐ͵݊݅ݏݐݏ݋‬െ ܿ‫ݐ͵݊݅ݏݐ͵ݏ݋‬ሽ
ʹ
ͳ ͳ
ൌ Ǥ ‫ܮ‬ሼʹܿ‫ ݐ͵݊݅ݏݐݏ݋‬െ ʹܿ‫ ݐ͵݊݅ݏݐ͵ݏ݋‬ሽ
ʹ ʹ
ͳ
ൌ ‫ܮ‬ሼ‫݊݅ݏ‬Ͷ‫ ݐ‬െ •‹ሺെʹ‫ݐ‬ሻ െ ‫݊݅ݏ‬͸‫ ݐ‬ሽ
Ͷ
ͳ
ൌ ‫ܮ‬ሼ‫݊݅ݏ‬Ͷ‫ ݐ‬൅ ‫ ݐʹ݊݅ݏ‬െ ‫݊݅ݏ‬͸‫ ݐ‬ሽ
Ͷ
ͳ Ͷ ʹ ͸
ൌ ቂ ൅ െ ቃ
Ͷ ‫ ʹ ݏ‬൅ͳ͸ ‫ ʹ ݏ‬൅Ͷ ‫ ʹ ݏ‬൅͵͸




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, λ ͵
2. If ‫ ݁ Ͳ׬‬െʹ‫‹• ݐ‬ሺ‫ ݐ‬൅ ߙ ሻ ‘•ሺ‫ ݐ‬െ ߙ ሻ ݀‫ ݐ‬ൌ , then find α
ͺ
[N14/CompIT/6M]
Solution:
By definition,
λ
‫ ݁ Ͳ׬‬െ‫݂ ݐݏ‬ሺ‫ݐ‬ሻ݀‫ ݐ‬ൌ ‫ ܮ‬ሼ݂ ሺ‫ݐ‬ሻሽ
Put ݂ ሺ‫ݐ‬ሻ ൌ •‹ሺ‫ ݐ‬൅ ߙ ሻ ‘•ሺ‫ ݐ‬െ ߙ ሻ
λ
‫ ݁ Ͳ׬‬െ‫‹• ݐݏ‬ሺ‫ ݐ‬൅ ߙ ሻ ‘•ሺ‫ ݐ‬െ ߙ ሻ ݀‫ ݐ‬ൌ ‫ܮ‬ሼ•‹ሺ‫ ݐ‬൅ ߙ ሻ ‘•ሺ‫ ݐ‬െ ߙ ሻሽ
ͳ
ൌ ‫ܮ‬ሼʹ •‹ሺ‫ ݐ‬൅ ߙ ሻ ‘•ሺ‫ ݐ‬െ ߙ ሻሽ
ʹ
ͳ
ൌ ‫ܮ‬ሼ•‹ሺʹ‫ݐ‬ሻ ൅ •‹ሺʹߙ ሻሽ
ʹ
λ െ‫ݐݏ‬ ͳ ʹ ‫ߙʹ ݊݅ݏ‬
‫׵‬ ‫Ͳ׬‬ ݁ •‹ሺ‫ ݐ‬൅ ߙ ሻ ‘•ሺ‫ ݐ‬െ ߙ ሻ ݀‫ ݐ‬ൌ ቂ ൅ ቃ
ʹ ‫ ʹ ݏ‬൅Ͷ ‫ݏ‬
Put ‫ ݏ‬ൌ ʹ,
λ ͳ ʹ ‫ߙʹ ݊݅ݏ‬
‫ ݁ Ͳ׬‬െʹ‫ݐ‬ •‹ሺ‫ ݐ‬൅ ߙ ሻ ‘•ሺ‫ ݐ‬െ ߙ ሻ ݀‫ ݐ‬ൌ ቂ ൅ ቃ
ʹ ͺ ʹ
͵ ͳ ͳ ‫ߙʹ ݊݅ݏ‬
ൌ ቂ ൅ ቃ
ͺ ʹ Ͷ ʹ
͵ ͳ ‫ߙʹ ݊݅ݏ‬
ൌ ൅
ͺ ͺ Ͷ
͵ ͳ ‫ߙʹ ݊݅ݏ‬
െ ൌ
ͺ ͺ Ͷ
ͳ ‫ߙʹ ݊݅ݏ‬
ൌ
Ͷ Ͷ
‫ ߙʹ݊݅ݏ‬ൌ ͳ
ߨ
ʹߙ ൌ
ʹ
ߨ
ߙൌ
Ͷ




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, 3. Find the Laplace Transform of ܿ‫ݐ͵ݏ݋ܿݐݏ݋ܿݐʹݏ݋‬
[N15/ChemBiot/5M]
Solution:
‫ܮ‬ሼܿ‫ݐ͵ݏ݋ܿݐݏ݋ܿݐʹݏ݋‬ሽ
ͳ
ൌ ‫ܮ‬ሼʹܿ‫ݐ͵ݏ݋ܿݐݏ݋ܿݐʹݏ݋‬ሽ
ʹ
ͳ
ൌ ‫ ܮ‬ሼሺܿ‫ ݐ͵ݏ݋‬൅ ܿ‫ݐݏ݋‬ሻܿ‫ݐ͵ݏ݋‬ሽ
ʹ
ͳ
ൌ ‫ܮ‬ሼ ‘• ʹ ͵‫ ݐ‬൅ ܿ‫ݐݏ݋ܿݐ͵ݏ݋‬ሽ
ʹ
ͳ ͳ
ൌ Ǥ ‫ܮ‬ሼʹ ‘• ʹ ͵‫ ݐ‬൅ ʹܿ‫ݐݏ݋ܿݐ͵ݏ݋‬ሽ
ʹ ʹ
ͳ
ൌ ‫ܮ‬ሼͳ ൅ ܿ‫ݏ݋‬͸‫ ݐ‬൅ ܿ‫ݏ݋‬Ͷ‫ ݐ‬൅ ܿ‫ݐʹݏ݋‬ሽ
Ͷ
ͳ ͳ ‫ݏ‬ ‫ݏ‬ ‫ݏ‬
ൌ ቂ ൅ ൅ ൅ ቃ
Ͷ ‫ݏ‬ ‫ ʹ ݏ‬൅͵͸ ‫ ʹ ݏ‬൅ͳ͸ ‫ ʹ ݏ‬൅Ͷ




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www.pinnacles.org.in Kalyan | Dombivli | Thane | Nerul | Dadar Contact - 9136008228