Quiz 9 (Solution)


Date: MATH 1210-4 - Spring 2004

  1. Evaluate $ \int (x^7+3\sqrt[3]{x}-\frac{3}{x^2}-8\sin(x))
dx$.

    \begin{displaymath}\begin{split}\int (x^7+3\sqrt[3]{x}-\frac{3}{x^2}-8\sin(x)) d...
...ac{9}{4} \sqrt[3]{x^4} + \frac{3}{x} + 8\cos(x) +C. \end{split}\end{displaymath}    

  2. Consider the equation $ \frac{dy}{dx} = x^2 y^2$.
    1. Find its general solution.

      We first rewrite $ \frac{dy}{dx} = x^2 y^2$ as $ \frac{1}{y^2} dy =
x^2 dx$ so that the variables are separated. Then we integrate both sides:

      \begin{displaymath}\begin{split}\int \frac{1}{y^2} dy &= \int x^2 dx\\ \frac{y^{...
...ac{x^3}{3} - C\\ y &= \frac{1}{-\frac{x^3}{3} - C}. \end{split}\end{displaymath}    

      Thus, the general solution is

      $\displaystyle y(x) = \frac{1}{-\frac{x^3}{3} -C}.$    

    2. Find the solution that satisfies $ y(0)=1$.

      We plug $ x=0$ into the general solution: $ 1 = y(0) =
\frac{1}{-\frac{0^3}{3} - C} = \frac{1}{-C}$, so that $ C=-1$ and the required solution is

      $\displaystyle y(x) = \frac{1}{-\frac{x^3}{3} +1}.$    



Javier Fernandez 2004-04-12