91Ó°ÊÓ

Suppose that \(\phi\) is a function of one variable. The differential equation \(d y / d x=\phi(y / x)\) is said to be homogeneous of degree \(0 .\) Let \(w(x)=y(x) / x .\) Differentiate both sides of the equation \(y(x)=x \cdot w(x)\) with respect to \(x .\) By equating the resulting expression for \(d y / d x\) with \(\phi(y / x),\) show that \(w(x)\) is the solution of a separable differential equation. Illustrate this theory by solving the differential equation \(d y / d x=2 x y /\left(x^{2}+y^{2}\right) .\) For this example, \(\phi(w)=2 w /\) \(\left(1+w^{2}\right)\)

Plot \(T(x)=4 x^{3}-3 x\) for \(-1 \leq x \leq 1\). Notice that the plot is contained in the square \([-1,1] \times[-1,1] .\) Of all degree 3 polynomials that have this containment property, \(T\) has the longest arc length. Use Simpson's Rule to calculate the arc length of the graph of \(y=T(x),-1 \leq x \leq 1\) to four decimal places of accuracy.

Suppose that \(\psi\) is a function of one variable and that \(a, b,\) and \(c\) are constants. If \(d y / d x=\psi(a+b x+c y),\) then show that \(u(x)=a+b x+c y(x)\) is the solution of a separable equation. Illustrate by solving the equation \(d y / d x=1+x+y\)

In each of Exercises 65-68, use the method of cylindrical shells to calculate the volume obtained by rotating the given planar region \(\mathcal{R}\) about the given line \(\ell\) \(\mathcal{R}\) is the region between the curves \(x=y^{3}\) and \(x=-y^{2} ; \ell\) is the line \(y=3\).

Find the center of mass of the given region \(\mathcal{R}\) \(\mathcal{R}\) is bounded above by \(y=4+2 x-x^{4}\) and below by \(y=x-1\)

In each of Exercises \(65-74\) calculate the expectation of a random variable whose probability density function is given. $$ e^{1-x} /(e-1) \quad 0 \leq x \leq 1 $$

The pressure \(P\) and temperature \(T\) in the outer envelope of a white dwarf (star) are related by the differential equation $$ \frac{d P}{d T}=C \frac{T^{7.5}}{P}, P(0)=0 $$ where \(C\) is a constant. Find \(P\) as a function of \(T\).

Find the center of mass of the given region \(\mathcal{R}\) \(\mathcal{R}\) is bounded above by \(y=\ln \left(1+x+x^{2}\right)\) and below by \(y=x^{2}-3\)

In each of Exercises 69-76, calculate the volume of the solid obtained when the region \(\mathcal{R}\) is rotated about the given line \(\ell\) \(\mathcal{R}\) is the region between \(y=6-x^{2}\) and \(y=-x ; \ell\) is the line \(x=-4\)

The surface of a flashlight reflector is obtained when \(y=2.05 \sqrt{x}+0.496,0.02 \leq x \leq 2.80 \mathrm{~cm}\) is rotated about the \(x\) -axis. Calculate its surface area to two significant digits.