Solving Differential Equations

Answer each question before opening its answer. This self-check does not record a quiz score.

1. To solve a first-order ODE you usually:

   • separate the variables
   • differentiate
   • use the quotient rule
   • guess
   Show answer and explanation

   Answer: separate the variables

   Then integrate.

2. For $\frac{dy}{dx} = xy$, you write:

   • $\int \tfrac{1}{y}\,dy = \int x\,dx$
   • $\int y\,dy = \int x\,dx$
   • $\int \tfrac{1}{x}\,dx = \int y\,dy$
   • $\int xy\,dx$
   Show answer and explanation

   Answer: $\int \tfrac{1}{y}\,dy = \int x\,dx$

   Separate.

3. The solution of $\frac{dy}{dx}=xy$ is:

   • $y = A e^{x^2/2}$
   • $y = A e^{x}$
   • $y = x^2/2 + c$
   • $y = \ln x$
   Show answer and explanation

   Answer: $y = A e^{x^2/2}$

   ln|y| = x²/2 + c.

4. The general solution contains:

   • an arbitrary constant
   • no constant
   • two variables
   • a derivative
   Show answer and explanation

   Answer: an arbitrary constant

   +c or A.

5. A boundary condition gives the:

   • particular solution
   • general solution
   • derivative
   • asymptote
   Show answer and explanation

   Answer: particular solution

   Fixes the constant.