91Ó°ÊÓ

Verify that the function is a solution of the differential equation on some interval, for any choice of the arbitrary constants appearing in the function. (a) \(y=c e^{2 x} ; \quad y^{\prime}=2 y\) (b) \(y=\frac{x^{2}}{3}+\frac{c}{x} ; \quad x y^{\prime}+y=x^{2}\) (c) \(y=\frac{1}{2}+c e^{-x^{2}} ; \quad y^{\prime}+2 x y=x\) (d) \(y=\left(1+c e^{-x^{2} / 2}\right) ;\left(1-c e^{-x^{2} / 2}\right)^{-1} \quad 2 y^{\prime}+x\left(y^{2}-1\right)=0\) (e) \(y=\tan \left(\frac{x^{3}}{3}+c\right) ; \quad y^{\prime}=x^{2}\left(1+y^{2}\right)\) (f) \(y=\left(c_{1}+c_{2} x\right) e^{x}+\sin x+x^{2} ; \quad y^{\prime \prime}-2 y^{\prime}+y=-2 \cos x+x^{2}-4 x+2\) (g) \(y=c_{1} e^{x}+c_{2} x+\frac{2}{x} ; \quad(1-x) y^{\prime \prime}+x y^{\prime}-y=4\left(1-x-x^{2}\right) x^{-3}\) (h) \(y=x^{-1 / 2}\left(c_{1} \sin x+c_{2} \cos x\right)+4 x+8 ;\) \(x^{2} y^{\prime \prime}+x y^{\prime}+\left(x^{2}-\frac{1}{4}\right) y=4 x^{3}+8 x^{2}+3 x-2\)

Suppose an object is launched from a point 320 feet above the earth with an initial velocity of 128 ft/sec upward, and the only force acting on it thereafter is gravity. Take \(g=32 \mathrm{ft} / \mathrm{sec}^{2}\). (a) Find the highest altitude attained by the object. (b) Determine how long it takes for the object to fall to the ground.

In Exercises \(12-22\) construct a direction field and plot some integral curves in the indicated rectangular region. $$ y^{\prime}=x y(y-1) ; \quad\\{-2 \leq x \leq 2,-4 \leq y \leq 4\\} $$