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Chapter 17

Differential Equations

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Problem 1

If $\frac{d y}{d x}=\frac{7 x^{2}}{y^{3}}$ and $y(3)=2,$ find an equation for $y$ in terms of $x$.

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Problem 2

If $\frac{d y}{d x}=5 x^{2} y$ and $y(0)=6,$ find an equation for $y$ in terms of $x$.

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Problem 3

If $\frac{d y}{d x}=\frac{e^{x}}{y^{2}}$ and $y(0)=1,$ find an equation for $y$ in terms of $x$.

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Problem 4

If $\frac{d y}{d x}=\frac{y^{2}}{x^{3}}$ and $y(1)=2,$ find an equation for $y$ in terms of $x$.

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Problem 5

If $\frac{d y}{d x}=\frac{\sin x}{\cos y}$ and $y(0)=\frac{3 \pi}{2},$ find an equation for $y$ in terms of $x$.

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Problem 6

A colony of bacteria grows exponentially and the colony's population is $4,000$ at time $t=0$ and $6,500$ at time $t=3 .$ How big is the population at time $t=10 ?$

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Problem 7

A rock is thrown upward with an initial velocity, $v(t),$ of 18 $\mathrm{m} / \mathrm{s}$ from a height, $h(t),$ of 45 $\mathrm{m}$ If the acceleration of the rock is a constant $-9 \mathrm{m} / \mathrm{s}^{2},$ find the height of the rock at time $t=4 .$

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Problem 8

A radioactive element decays exponentially in proportion to its mass. One-half of its original amount remains after $5,750$ years. If $10,000$ grams of the element are present initially, how much will be left after $1,000$ years?

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Problem 9

Sketch the slope field for $\frac{d y}{d x}=2 x$.

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Problem 10

Sketch the slope field for $\frac{d y}{d x}=-\frac{x}{y}$.

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