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Chapter 7: Problem 4
Is a reduction formula an analytical method or a numerical method? Explain.
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Consider the general first-order initial value problem \(y^{\prime}(t)=a y+b, y(0)=y_{0},\) for \(t \geq 0,\) where \(a, b,\) and \(y_{0}\) are real numbers. a. Explain why \(y=-b / a\) is an equilibrium solution and corresponds to horizontal line segments in the direction field. b. Draw a representative direction field in the case that \(a>0\). Show that if \(y_{0}>-b / a,\) then the solution increases for \(t \geq 0\) and if \(y_{0}<-b / a,\) then the solution decreases for \(t \geq 0\). c. Draw a representative direction field in the case that \(a<0\). Show that if \(y_{0}>-b / a,\) then the solution decreases for \(t \geq 0\) and if \(y_{0}<-b / a,\) then the solution increases for \(t \geq 0\).
Let \(a>0\) and \(b\) be real numbers. Use integration to confirm the following identities. (See Exercise 68 of Section 7.2) a. \(\int_{0}^{\infty} e^{-a x} \cos b x d x=\frac{a}{a^{2}+b^{2}}\) b. \(\int_{0}^{\infty} e^{-a x} \sin b x d x=\frac{b}{a^{2}+b^{2}}\)
Consider the family of functions \(f(x)=1 / x^{p},\) where \(p\) is a real number. For what values of \(p\) does the integral \(\int_{0}^{1} f(x) d x\) exist? What is its value?
\(A\) powerful tool in solving problems in engineering and physics is the Laplace transform. Given a function \(f(t)\) the Laplace transform is a new function \(F(s)\) defined by $$F(s)=\int_{0}^{\infty} e^{-s t} f(t) d t$$ where we assume that \(s\) is a positive real number. For example, to find the Laplace transform of \(f(t)=e^{-t},\) the following improper integral is evaluated: $$F(s)=\int_{0}^{\infty} e^{-s t} e^{-t} d t=\int_{0}^{\infty} e^{-(s+1) t} d t=\frac{1}{s+1}$$ Verify the following Laplace transforms, where a is a real number. $$f(t)=e^{a t} \longrightarrow F(s)=\frac{1}{s-a}$$
Use integration by parts to evaluate the following integrals. $$\int_{0}^{\infty} x e^{-x} d x$$
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