Split, integrate, fix the constant

Mathematics · Differential Equations · ages 19-20

Name ______________________ Date ____________

Which of these equations cannot be separated into a function of x times a function of y?
- dy/dx equals x plus y
- dy/dx equals x times y
- dy/dx equals y divided by x
- dy/dx equals x squared
Solve dy/dx equals 2x with y of 0 equals 3. What is y of 1?
Answer: ______________
Solve dy/dx equals 2y over x with y of 1 equals 1, for x greater than 0. What is y of 3?
Answer: ______________
Which of these equations cannot be separated into an x-piece times a y-piece?
- dy/dx = x y
- dy/dx = y over x
- dy/dx = x + y
After separating and integrating, you may drop the constant because the initial condition replaces it.
Circle one: True False
Solve dy/dx = x y with y(0) = 2. Which is the solution?
- y = 2 + x squared over 2
- y = 2 e to the x squared over 2
- y = e to the 2x
Dividing dy/dx = x y by y loses which solution?
- y = 1
- y = x
- y = 0
Solve dy/dx = 2y over x with y(1) = 1, for x greater than 0. What is y(3)?
Answer: ______________
Lee integrates dy/dx = 2x to y = x squared, plugs in y(0) = 3, and reports no solution. What went wrong?
- Lee integrated the wrong side
- Lee dropped the constant C
- Lee used x = 1 too soon
For dy/dx = 2y over x with y(3) = 9, where does the solution hold?
- For x greater than 0 only
- For every x
- For x less than 0 only

Answer key

For grown-ups. Fold this page away before handing over the rest.

dy/dx equals x plus y · The sum x plus y never splits into a product of one factor in x and one in y.
4 · Integrating gives x squared plus C, C is 3, and at 1 that is 4.
9 · Separation gives y equals A times x squared, A is 1, and 3 squared is 9.
dy/dx = x + y · A sum of x and y never factors into a product.
False · C carries the whole family; the condition only fixes its value.
y = 2 e to the x squared over 2 · Separating gives ln|y| = x squared over 2 + C, and the condition sets C.
y = 0 · y = 0 vanishes in the division, yet it satisfies the equation.
9 · Separating gives y = C x squared, C is 1, and at 3 that is 9.
Lee dropped the constant C · With C kept, y = x squared + 3 fits the condition at once.
For x greater than 0 only · Dividing by x breaks at 0, so the curve cannot cross it.