Quiz 9 (Solution)

Date: MATH 1210-4 - Spring 2004

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

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

   
   

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:

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

      
      

      Thus, the general solution is

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

      
      

   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

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