Chapter 17

Derive Johann Bernoulli's differential equation for the catenary, \(d y / d x=s / a\), as follows: Let the lowest point of the hanging cord be the origin of the coordinate system, and consider a piece of the chord of length \(s\) over the interval into a closed-form expression. \([0, x]\). Let \(T(x)\) be the (vector) tension of the cord at the point \(P=(x, y)\). Let \(\alpha\) be the angle that \(T(x)\) makes with 3. Derive Johann Bernoulli's differential equation for the the horizontal and let \(\rho\) be the density of the cord. Show that the equilibrium of horizontal forces gives the equation. equation \(d s=\frac{\sqrt{a} d y}{\sqrt{x}}\) \(|T(0)|=|T(x)| \cos \alpha\), while that of the vertical forces gives \(\rho s=|T(x)| \sin \alpha .\) Since \(d y / d x=\tan \alpha\), Bernoulli's equaDapro tion can be derived by dividing the second equation by the differences of the parts is equal to the difference of the sums first. of the parts.

. Suppose \(A=a_{1}+a_{2}+\cdots+a_{n}\) and \(B=b_{1}+b_{2}+\cdots+\) \(b_{n} .\) Show that \(\sum\left(b_{i}-a_{i}\right)=B-A\), or, that the sum of the differences of the parts is equal to the difference of the sums of the parts.

Translate Leibniz's solution of \(m d x+n y d x+d y=0\) into modern terms by noting that \(d p / p=n d x\) is equiv-

The modern way to derive Kepler's area law is to break the force into its radial and transverse components, rather than the tangential and normal components used by Hermann, and use polar coordinates whose origin is the center of force. Assume then that the center of force is at the origin of a polar coordinate system. Using vector notation, \(\operatorname{set} \mathbf{u}_{r}=\mathbf{i} \cos \theta+\mathbf{j} \sin \theta\) and \(\mathbf{u}_{\theta}=-\mathbf{i} \sin \theta+j \cos \theta .\) Show that \(d \mathbf{u}_{r} / d \theta=\mathbf{u}_{\theta}\) and \(d \mathbf{u}_{\theta} / d \theta=-\mathbf{u}_{r}\). Then show that if \(\mathbf{r}=r \mathbf{u}_{r}\), then the velocity \(\mathbf{v}\) is given by \(r(d \theta / d t) \mathbf{u}_{\theta}+\) \((d r / d t) \mathbf{u}_{r}\). Show next that the radial component \(a_{r}\) and the transverse component \(a_{\theta}\) of the acceleration are given by $$ a_{r}=\frac{d^{2} r}{d t^{2}}-r\left(\frac{d \theta}{d t}\right)^{2} \quad \text { and } \quad a_{\theta}=r \frac{d^{2} \theta}{d t^{2}}+2 \frac{d r}{d t} \frac{d \theta}{d t} $$ Since the force is central, \(a_{\theta}=0 .\) Multiply the differential equation expressing that fact by \(r\) and integrate to get \(r^{2} \frac{d \theta}{d t}=k\), where \(k\) is a constant. Show finally that \(r^{2} \frac{d \theta}{d t}=\) \(\frac{d A}{d t}\), where \(A\) is the area swept out by the radius vector. This proves Kepler's law of areas.

Show that the differential equation $$ \frac{d^{2} r}{d t^{2}}-r\left(\frac{d \theta}{d t}\right)^{2}=-\frac{k}{r^{2}} $$ derived by assuming that the component \(a_{r}\) of the force from Exercise 6 is inversely proportional to \(r^{2}\) is equivalent to the differential equation Hermann derived using the inverse square property of the central force.

Show that the equation \(a \pm c x / b=\sqrt{x^{2}+y^{2}}\) is a parabola if \(b=c\), is an ellipse if \(b>c\), and is a hyperbola if \(b

Show that \(y=e^{x / a}\) is a solution to the differential equation \(a^{3} d^{3} y-y d x^{3}=0\). Next, assume that the product \(e^{-(x / a)}\left(a^{3} d^{3} y-y d x^{3}\right)\) is the differential of $$ e^{-(x / a)}\left(A d^{2} y+B d y d x+C y d x^{2}\right) $$

. Show that if \(y=u e^{\alpha x}\) is assumed to be a solution of \(a^{2} d^{2} y+a d y d x+y d x^{2}=0\), then if \(\alpha=-1 / 2 a\), conclude that \(u\) is a solution to \(a^{2} d^{2} u+(3 / 4) u d x^{2}=0\).

16\. Solve \(a^{2} d^{2} u+(3 / 4) u d x^{2}=0 .\) First multiply by \(d u\) and integrate once to get \(4 a^{2} d u^{2}=\left(K^{2}-3 u^{2}\right) d x^{2}\) or $$ d x=\frac{2 a}{\sqrt{K^{2}-3 u^{2}}} d u $$ Integrate a second time to get $$ x=\frac{2 a}{\sqrt{3}} \arcsin \frac{\sqrt{3} u}{K}-f $$ Rewrite this equation for \(u\) in terms of \(x\) as $$ u=C \sin \left(\frac{(x+f) \sqrt{3}}{2 a}\right) $$

17\. Find the natural logarithms of the three cube roots of 1 and of the five fifth roots of 1 .

18\. Find the curve joining two points in the upper half-plane, which, when revolved around the \(x\) axis, generates a surface of minimal surface area. If \(y=f(x)\) is the equation of the curve, then the desired surface area is \(I=\) \(2 \pi \int y d s=2 \pi \int y \sqrt{1+y^{2}} d x .\) So use the Euler equation in the modified form \(F-y^{\prime}\left(\partial F / \partial y^{\prime}\right)=c\), where \(F=y \sqrt{1+y^{\prime 2}} .\) (Hint: Begin by multiplying the equation through by \(\sqrt{1+y^{\prime 2}}\).)

Determine a procedure for finding the differential equation of a family of orthogonal trajectories to a given family \(f(x, y, \alpha)=0\). (Use the fact that orthogonal lines have negative reciprocal slopes.) Use your procedure to find the family orthogonal to the family of hyperbolas \(x^{2}-y^{2}=a^{2}\). 20\. Determine and solve the differential equation for the family of synchrones, the family orthogonal to the family of brachistochrones.

21\. Solve the differential equation \(\left(2 x y^{3}+6 x^{2} y^{2}+8 x\right) d x+\) \(\left(3 x^{2} y^{2}+4 x^{3} y+3\right) d y=0\) using Clairaut's method.

22\. Use Clairaut's technique of multiple integration to calculate the volume of the solid bounded by the cylinders \(a x=\) \(y^{2}, b y=z^{2}\) and the coordinate planes. First determine the volume element \(d x \int z d y\) by converting the integrand to a function of \(x\) and integrating. Then integrate the volume element with appropriate limits. Compare this method to the standard modern method.

23\. Suppose that \(x\) and \(y\) are given in terms of \(t\) and \(u\) by the functions $$ x=\frac{t}{\sqrt{1+u^{2}}}, \quad y=\frac{t u}{\sqrt{1+u^{2}}} $$ Show that the change-of-variable formula is given by $$ d x d y=\frac{t d t d u}{1+u^{2}} $$

24\. Suppose that the solution to the wave equation \(\frac{2^{2} y}{\partial t^{2}}=\frac{\partial^{2} y}{\partial x^{2}}\) is given by \(y=\Psi(t+x)-\Psi(t-x)\). Show that the initial conditions \(y(0, x)=f(x), y^{\prime}(0, x)=g(x)\) and the condition \(y(t, 0)=y(t, l)=0\) for all \(t\) lead to the requirements that \(f(x)\) and \(g(x)\) are odd functions of period \(2 l\) (d'Alembert).

25\. Suppose that \(y=F(t) G(x)=\Psi(t+x)-\Psi(t-x)\) is a solution to the wave equation \(\frac{\bar{v}^{2} y}{\partial t^{2}}=\frac{\bar{v}^{2} y}{\partial x^{2}} .\) Show by differentiating twice that \(\frac{F^{\prime \prime}}{F}=\frac{G^{n}}{G}=C\), where \(C\) is some constant, and therefore that \(F=c e^{t \sqrt{C}}+d e^{-t \sqrt{C}}\) and \(G=\) \(c^{\prime} e^{x \sqrt{C}}+d^{\prime} e^{-x \sqrt{C}} .\) Apply the condition \(y(t, 0)=y(t, l)=\) 0 to show that \(C\) must be negative, and hence derive the solution \(F(t)=A \cos N t, G(x)=B \sin N x\) for the appropriate choice of \(A, B\), and \(N\) (d'Alembert).

26\. Find the isosceles triangle of smallest area that circumscribes a circle of radius 1 (Simpson). (Simpson). mum when \(x=\frac{1}{2} b \sqrt[3]{5}, y=\frac{1}{4} b \sqrt[3]{5}\), and \(z=\frac{b \sqrt[3]{5}}{2 \sqrt{3}}\) (Simpson).

27\. Find the cone of least surface area with given volume \(V\) (Simpson).

28\. Show that \(w=\left(b^{3}-x^{3}\right)\left(x^{2} z-z^{3}\right)\left(x y-y^{2}\right)\) has a maximum when \(x=\frac{1}{2} b \sqrt[3]{5}, y=\frac{1}{4} b \sqrt[3]{5}\), and \(z=\frac{b \sqrt[3]{5}}{2 \sqrt{3}}\) (Simpson).

29\. Calculate the first four nonzero terms of the power series for \(y=\cos z\) using Maclaurin's technique without explicitly using the derivatives of the cosine or sine. Assume that the radius of the circle is 1 .

32\. Sketch a particular example of the "witch of Agnesi," the curve given by \(y^{2}=\frac{4(2-x)}{x}\). Show that it is symmetric about

33\. Assume that after the flood the human population was 6 and that 200 years later the population was \(1,000,000\). Find the annual rate of growth of the population (Euler). (Hint: If the annual rate of growth is \(1 / x\), then the equation for the problem is $$ \left.6\left(\frac{1+x}{x}\right)^{200}=1,000,000 .\right) $$

36\. Replace \(x\) by \(i x\) in the expansion in the text for \(\left(e^{x}-\right.\) \(\left.e^{-x}\right) / 2\) to get both the power series for the sine and a representation of the sine as an infinite product. By using the relationship between the roots and coefficients of a polynomial (extended to power series), show that $$ \sum_{k=1}^{\infty} \frac{1}{k^{2}}=\frac{\pi^{2}}{6} \text { and } \sum_{k=1}^{\infty} \frac{1}{k^{4}}=\frac{\pi^{4}}{90} $$

37\. Determine all the relative extrema for \(V=x^{3}+y^{2}-\) \(3 x y+(3 / 2) x\), and for each one determine whether it is a maximum or minimum. Compare your answer with that of Euler.

38\. If \(y=\arctan x\), show that \(\sin y=x / \sqrt{1+x^{2}}\) and \(\cos y=\) \(1 / \sqrt{1+x^{2}}\). Then, if \(p=x / \sqrt{1+x^{2}}\), show that \(\sqrt{1-p^{2}}=\) \(1 / \sqrt{1+x^{2}}\). Since \(y=\arcsin p\), it follows that \(d y=\) \(d p / \sqrt{1-p^{2}}\) and \(d p=d x /\left(1+x^{2}\right)^{3 / 2}\). Conclude that $$ d y=\frac{d x}{1+x^{2}} \quad \text { (Euler) } $$

39\. Calculate \(d y\) for \(y=a^{x}\) by noting that \(d y=a^{x+d x}-a^{x}=\) \(a^{x}\left(a^{d x}-1\right)\) and then expanding \(a^{d x}-1\) into the power series \(\ln a d x+\frac{(\ln a)^{2} d x^{2}}{2}+\cdots\) (Euler).

44\. Use Lagrange's technique to calculate the quantities \(p, q\), \(r\) for the function \(f(x)=\sqrt{x}\) and thus determine the first. three terms of its power series representation.

. Show why Lagrange's power series representation fails for the case \(f(x)=e^{-1 / x^{2}}\).

. Given that \(f(x+i)=f(x)+p i+q i^{2}+r i^{3}+\cdots\), show that \(p=f^{\prime}(x), q=f^{\prime \prime}(x) / 2 !, r=f^{m \prime \prime}(x) / 3 !, \ldots\)

Did eighteenth-century mathematicians prove or use the fundamental theorem of calculus in the sense it is used today? What concepts must be defined before one can even consider this theorem? How are these concepts dealt with by eighteenth-century mathematicians? Did these mathematicians consider the fundamental theorem as "fundamental"?