
# 3.E: Chapter Review Excercises


## Exercise $$\PageIndex{1}$$

Solve the following differential equations:

1. $$\displaystyle \frac{dy}{dx}=\frac{2 \sqrt{1+y^2}}{x}$$
2. $$\displaystyle \frac{dy}{dx}=\frac{2xy}{x^2+4x+5}$$
3. $$\displaystyle x \frac{dv}{dx}=\frac{1-v^2}{2v}$$
4. $$(x+1)\displaystyle \frac{dy}{dx} +y= F(x),$$ where

$F(x)= \left\{\begin{array}{c} x,0\leq x <3,\\ 3,x \geq 3, \end{array} \right. y(0)= \frac{1}{2}.$

5. $$y^2(1-x^2)^{1/2}dy=sin^{-1} x dx, \, y(0)=0$$

6. $$\displaystyle \frac{dy}{dx}=e^{4x+3y}.$$

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## Exercise $$\PageIndex{2}$$

Newton's law of cooling states that a hot object introduced into a cooler environment will cool at a rate proportional to the excess of its temperature above that of its environment.

If a cup of coffee sitting in a room maintained at a temperature of $$20^\circ C$$ cools from $$80^\circ C$$ to $$50^\circ C$$ in $$5$$min, how much longer will it take to cool to $$40^\circ C$$?

## Exercise $$\PageIndex{3}$$
A large tank initially contains $$50$$ gal of brine in which there is dissolved $$10$$ lb of salt. Brine containing $$2$$ lb of salt per gallon flows into the tank at a rate of $$5 gal/min$$. the mixture is kept uniform by stirring, and the stirred mixture simultaneously flows out at the rate of $$3 gal/min$$. How much salt is in the tank at any time $$t>0?$$