Answer: First, separate the variables: = 4x dx. Then, take the integral of both sides.
∫ = ∫^4 x dx
Next, integrate both sides: − = 2x^2 + C. You can rewrite this as y = –
Finally, solve for the constant.
1 = , so C = − 1The solution is y = .
PROBLEM 3. If = and y(1) = , find an equation for y in terms of x.
Answer: This time, separating the variables gives us this: = .
Then, take the integral of both sides:∫ = ∫.
Next, integrate both sides.
− = ln x + CAnd rearrange the equation.
Finally, solve for the constant. = , so C = −3. The solution is y = .
PROBLEM 4. A city had a population of 10,000 in 1980 and 13,000 in 1990. Assuming an exponential
growth rate, estimate the city’s population in 2000.
Answer: The phrase “exponential growth rate” means that = ky, where k is a constant. Take the
integral of both sides.