3.E: Chapter Review Excercises
- Page ID
- 25897
Exercise \(\PageIndex{1}\)
Solve the following differential equations:
- \(\displaystyle \frac{dy}{dx}=\frac{2 \sqrt{1+y^2}}{x}\)
- \(\displaystyle \frac{dy}{dx}=\frac{2xy}{x^2+4x+5}\)
- \(\displaystyle x \frac{dv}{dx}=\frac{1-v^2}{2v}\)
- \((x+1)\displaystyle \frac{dy}{dx} +y= F(x),\) where
\[ F(x)= \left\{\begin{array}{c}
x,0\leq x <3,\\
3,x \geq 3,
\end{array}
\right.
y(0)= \frac{1}{2}.\]
- \(y^2(1-x^2)^{1/2}dy=sin^{-1} x dx, \, y(0)=0\)
- \(\displaystyle \frac{dy}{dx}=e^{4x+3y}.\)
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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 \(40^\circ C \)?
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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 \(t>0?\)
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