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Chapter 22: Problem 10

Given \(\mathcal{L}\\{x\\}=X, x(0)=2, x^{\prime}(0)=3\) find the Laplace transform of each of the following expressions: (a) \(x^{\prime}\) (b) \(x^{\prime \prime}\) (c) \(2 x^{\prime}-x\) (d) \(3 x^{\prime \prime}+2 x^{\prime}-x\) (e) \(-2 x^{\prime \prime}+x^{\prime}+3 x\)

Short Answer

Expert verified
a. x'(t) b. x''(t) c. 2x'(t) - x(t) d. 3x''(t) + 2x'(t) - x(t) e. -2x''(t) + x'(t) + 3x(t) Answer: a. The Laplace transform of x'(t) is \(sX(s)-2\). b. The Laplace transform of x''(t) is \(s^2X(s)-2s-3\). c. The Laplace transform of 2x'(t) - x(t) is \(2sX(s) - 4 - X(s)\). d. The Laplace transform of 3x''(t) + 2x'(t) - x(t) is \(3s^2X(s) - 6s - 9 + 2sX(s) - 4 - X(s)\). e. The Laplace transform of -2x''(t) + x'(t) + 3x(t) is \(-2s^2X(s) + 4s + 6 + sX(s) - 2 + 3X(s)\).

Step by step solution

01

Find the Laplace Transform of x'(t) (Expression a)

We're given that \(\mathcal{L}\\{x(t)\\} = X(s)\). Applying the Laplace transform to the derivative x'(t), we have: $$ \mathcal{L}\\{x'(t)\\} = sX(s) - x(0) $$ Now, substitute the given initial condition x(0) = 2: $$ \mathcal{L}\\{x'(t)\\} = sX(s) -2 $$ So, the Laplace transform of x'(t) is \(sX(s)-2\).
02

Find the Laplace Transform of x''(t) (Expression b)

Applying the Laplace transform to the second derivative x''(t), we have: $$ \mathcal{L}\\{x''(t)\\} = s^2X(s) - sx(0) - x'(0) $$ Now, substitute the given initial conditions x(0) =2 and x'(0) =3: $$ \mathcal{L}\\{x''(t)\\} = s^2X(s) - 2s - 3 $$ So, the Laplace transform of x''(t) is \(s^2X(s)-2s-3\).
03

Find the Laplace Transform of 2x'-x (Expression c)

Now, we need to find the Laplace transform of the expression 2x'(t) - x(t): $$ \mathcal{L}\\{2x'(t) - x(t)\\} = 2\mathcal{L}\\{x'(t)\\} - \mathcal{L}\\{x(t)\\} $$ Substitute the Laplace transforms we found in steps 1 and 2, and the given Laplace transform of \(x(t)\) as \(X(s)\): $$ \mathcal{L}\\{2x'(t) - x(t)\\} = 2(sX(s)-2) - X(s) $$ So, the Laplace transform of 2x'(t) - x(t) is \(2sX(s) - 4 - X(s)\).
04

Find the Laplace Transform of 3x''+2x'-x (Expression d)

Now, we need to find the Laplace transform of the expression 3x''(t) + 2x'(t) - x(t): $$ \mathcal{L}\\{3x''(t) + 2x'(t) - x(t)\\} = 3\mathcal{L}\\{x''(t)\\} + 2\mathcal{L}\\{x'(t)\\} - \mathcal{L}\\{x(t)\\} $$ Substitute the Laplace transforms we found in steps 1, 2, and the given Laplace transform of x(t) as X(s): $$ \mathcal{L}\\{3x''(t) + 2x'(t) - x(t)\\} = 3(s^2X(s) - 2s - 3) + 2(sX(s)-2) - X(s) $$ So, the Laplace transform of 3x''(t) + 2x'(t) - x(t) is \(3s^2X(s) - 6s - 9 + 2sX(s) - 4 - X(s)\).
05

Find the Laplace Transform of -2x''+x'+3x (Expression e)

Now, we need to find the Laplace transform of the expression -2x''(t) + x'(t) + 3x(t): $$ \mathcal{L}\\{-2x''(t) + x'(t) + 3x(t)\\} = -2\mathcal{L}\\{x''(t)\\} + \mathcal{L}\\{x'(t)\\} + 3\mathcal{L}\\{x(t)\\} $$ Substitute the Laplace transforms we found in steps 1, 2, and the given Laplace transform of x(t) as X(s): $$ \mathcal{L}\\{-2x''(t) + x'(t) + 3x(t)\\} = -2(s^2X(s) - 2s - 3) + (sX(s)-2) + 3X(s) $$ So, the Laplace transform of -2x''(t) + x'(t) + 3x(t) is \(-2s^2X(s) + 4s + 6 + sX(s) - 2 + 3X(s)\).

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Most popular questions from this chapter

Solve the following differential equations using the Laplace transform method: (a) \(x^{\prime}-x=\mathrm{e}^{t}, x(0)=0\) (b) \(x^{n \prime}+x^{\prime}+2 x=4 \cos 2 t, x(0)=-1\), \(x^{\prime}(0)=2\) (c) \(x^{\prime \prime}+x^{\prime}-6 x=1+7 t-6 t^{3}\) \(x(0)=x^{\prime}(0)=0\) (d) \(x^{\prime \prime}-4 x=4 t, x(0)=x^{\prime}(0)=1\) (e) \(x^{\prime \prime}+x=2 \sin t, x(0)=x^{\prime}(0)=0\)

The second shift theorem states that if \(\mathcal{L}\\{f(t)\\}=F(s)\) then \(\mathcal{L}\\{u(t-d) f(t-d)\\}=\mathrm{e}^{-s d} F(s), \quad d>0\) where \(u(t)\) is the unit step function. (a) Prove this theorem. (b) Find the Laplace transform of \(u(t-3)(t-3)^{5}\) (c) Find the inverse Laplace transform of $$ \frac{\mathrm{e}^{-4 s} 4 !}{s^{5}} $$

Find the inverse Laplace transform of each of the following: (a) \(\frac{4}{s^{2}}\) (b) \(-\frac{3}{s}\) (c) \(\frac{5}{s^{3}}\) (d) \(\frac{3}{s^{5}}-\frac{2}{s^{4}}\) (e) \(\frac{1}{3 s^{5}}-\frac{2}{3 s^{2}}\)

The Laplace transform of \(y(t)\) is \(Y(s), y(0)=2\) and \(y^{\prime}(0)=-1\). Find the Laplace transform of the following expressions: (a) \(y^{\prime}\) (b) \(y^{\prime \prime}\) (c) \(2 y^{\prime \prime}-y^{\prime}+3 y\) (d) \(-y^{\prime \prime}+4 y^{\prime}-6 y\) (e) \(\frac{y^{\prime \prime}}{2}+3 y^{\prime}-y\)

Find the inverse Laplace transform of (a) \(\frac{1}{2(s+3)}\) (b) \(\frac{1}{2 s+3}\) (c) \(\frac{3 s}{s^{2}+1}\) (d) \(\frac{-6}{s^{2}+9}\) (e) \(\frac{s+2}{s^{2}+4}\)
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