# Laplace transform questions and answers

Recent questions in Laplace transform
Laplace transform

### Use Laplace transform to solve the following initial value problem: $$\displaystyle y''+{5}{y}={1}+{t},{y}{\left({0}\right)}={0},{y}'{\left({0}\right)}={4}$$ A)$$\displaystyle{\frac{{{7}}}{{{25}}}}{e}^{{t}}{\cos{{\left({2}{t}\right)}}}+{\frac{{{21}}}{{{10}}}}{e}^{{t}}{\sin{{\left({2}{t}\right)}}}$$ B) $$\displaystyle{\frac{{{7}}}{{{2}}}}+{\frac{{{t}}}{{{5}}}}-{\frac{{{t}}}{{{25}}}}{e}^{{t}}{\cos{{\left({2}{t}\right)}}}+{\frac{{{21}}}{{{10}}}}{e}^{{t}}{\sin{{\left({2}{t}\right)}}}$$ C) $$\displaystyle{\frac{{{7}}}{{{25}}}}+{\frac{{{t}}}{{{5}}}}$$ D) $$\displaystyle{\frac{{{7}}}{{{25}}}}+{\frac{{{t}}}{{{5}}}}-{\frac{{{7}}}{{{25}}}}{e}^{{t}}{\cos{{\left({2}{t}\right)}}}+{\frac{{{21}}}{{{10}}}}{e}^{{t}}{\sin{{\left({2}{t}\right)}}}$$ E) non of the above F) $$\displaystyle{\frac{{{7}}}{{{25}}}}+{\frac{{{t}}}{{{5}}}}-{\frac{{{7}}}{{{5}}}}{e}^{{t}}{\cos{{\left({2}{t}\right)}}}+{\frac{{{21}}}{{{10}}}}{e}^{{t}}{\sin{{\left({2}{t}\right)}}}$$

Laplace transform

### Find the inverse Laplace transform of $$\displaystyle{F}{\left({s}\right)}={\frac{{{7}{s}-{6}}}{{{s}^{{2}}-{4}}}}$$

Laplace transform

### find the Laplace transform of f (t). $$\displaystyle{f{{\left({t}\right)}}}={t}^{{2}}{\cos{{2}}}{t}$$

Laplace transform

### Let f(t) be a function on $$\displaystyle{\left[{0},\infty\right]}$$. The Laplace transform of f is the function F defined by the integral $$\displaystyle{F}{\left({s}\right)}={\int_{{0}}^{{\infty}}}{e}^{{-{s}{t}}}{f{{\left({t}\right)}}}{\left.{d}{t}\right.}$$ . Use this definition to determine the Laplace transform of the following function. $$\displaystyle{f{{\left({t}\right)}}}={e}^{{-{4}{t}}}{\sin{{9}}}{t}$$

Laplace transform

### Find the laplace transform $$\displaystyle{L}{\left[{\left({t}^{{2}}+{4}\right)}{e}^{{{2}{t}}}-{e}^{{-{2}{t}}}{\cos{{t}}}\right]}$$

Laplace transform

### determine the Laplace transform of f. $$\displaystyle{f{{\left({t}\right)}}}={e}^{{-{2}{t}}}{\sin{{\left({t}-{\frac{{\pi}}{{{4}}}}\right)}}}$$

Laplace transform

### Find Laplace transformation of the intengral from 0 to t of $$\displaystyle\tau{\cos{{\left({3}\tau\right)}}}{d}\tau$$

Laplace transform

### find the inverse Laplace transform of the given function. $$\displaystyle{F}{\left({s}\right)}={\frac{{{3}!}}{{{\left({s}-{2}\right)}^{{4}}}}}$$

Laplace transform

### Find the inverse Laplace transform of the following: $$\displaystyle{H}{\left({s}\right)}={\frac{{{2}{e}^{{-{s}}}}}{{{s}^{{2}}{\left({s}+{3}\right)}}}}$$

Laplace transform

### Find the inverse Laplace transform of the following transfer function: $$\displaystyle{\frac{{{Y}{\left({s}\right)}}}{{{U}{\left({s}\right)}}}}={\frac{{{50}}}{{{\left({s}+{7}\right)}^{{2}}+{25}}}}$$ Select one: a)$$\displaystyle{f{{\left({t}\right)}}}={10}{e}^{{-{7}{t}}}{\sin{{\left({5}{t}\right)}}}$$ b)$$\displaystyle{f{{\left({t}\right)}}}={10}{e}^{{{7}{t}}}{\sin{{\left({5}{t}\right)}}}$$ c)$$\displaystyle{f{{\left({t}\right)}}}={50}{e}^{{-{7}{t}}}{\sin{{\left({5}{t}\right)}}}$$ d)$$\displaystyle{f{{\left({t}\right)}}}={2}{e}^{{-{7}{t}}}{\sin{{\left({5}{t}\right)}}}$$

Laplace transform

### Do the laplace transform of $$L\left\{t-e^{3t}\right\}$$

Laplace transform

### Derive Laplace Transform of the following: a)$$\displaystyle{f{{\left({t}\right)}}}={e}^{{-{a}{t}}}$$ b) $$f(t)=t$$c) $$\displaystyle{f{{\left({t}\right)}}}={t}^{{2}}$$ d) $$\displaystyle{\cosh}{t}$$

Laplace transform

### to find the inverse Laplace transform of the given function. $$\displaystyle{F}{\left({s}\right)}={\frac{{{2}^{{{n}+{1}}}{n}!}}{{{s}^{{{n}+{1}}}}}}$$

Laplace transform

### Compute the Laplace transform of $$\displaystyle{f{{\left({t}\right)}}}={\cos{{\left({R}{t}-{7}\right)}}}$$ take R as 70

Laplace transform

### determine the inverse Laplace transform of the function. $$L^{-1}\left\{R(s)\right\}=L^{-1}\left\{\frac{7}{(s+3)(s-3)}\right\}$$

Laplace transform

### Find Laplace transforms of $$\displaystyle{\sin{{h}}}{3}{t}$$ $$\displaystyle{{\cos}^{{2}}{2}}{t}$$

Laplace transform

### Use Laplace transforms to solve the initial value problems. $$\displaystyle{y}\text{}{y}'=\delta{\left({t}\right)}-\delta{\left({t}-{3}\right)},{y}{\left({0}\right)}={1},{y}'{\left({0}\right)}={0}$$

Laplace transform

### determine the inverse Laplace transform of the given function. $$\displaystyle{F}{\left({s}\right)}={\frac{{{2}{s}+{1}}}{{{s}^{{2}}+{16}}}}$$

Laplace transform