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Mathematics LibreTexts

3.E: Chapter Review Excercises

  • Page ID
    25897
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    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. \(\displaystyle \frac{dy}{dx}=e^{4x+3y}.\)

    Answer

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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 \(40^\circ C \)?

    Answer

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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 \(t>0?\)

    Answer

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