Download differential equations project

A differential equation is used to find the rate at which something is changing at a certain point.
It is used to find an unknown function based on the dy/dx equation that is given as well as a certain x
and y value.
Steps
1.
2.
3.
4.
Separate Variables
Integrate both sides (don’t forget the “+C”)
Substitute and solve for “C”
Plug in “C” and solve for y
Example 1
Find y=f(x) when dy/dx=x2y2, and x=1 when y=1
Step 1
1
dy =x2dx or y-2dy=x2dx
2
y
Step 2
∫y-2dy=∫x2dx
Step 3
(1,1)
-(1)-1=⅓ (1)3+C
-1=⅓+C
C=-4/3
Step 4
-y-1=⅓x3-4/3
y-1=-⅓x3+4/3
y=(-⅓x3+4/3)-1
-y-1=⅓x3+C
Example 2
Find y=f(x) when dy/dx=-x/y, with the point (0,5)
Step 1
ydy=xdx
Step 2
∫ydy=∫xdx
Step 3
(0,5)
½(5)2=½ (0)2+C
½(25)=0+C
12.5=C
Step 4
½y2=½x2+12.5
y2=x2+25
y=√x2+25
½y2=½x2+C
Practice Problems
Non-Calculator
Multiple Choice
1) What is the particular solution of the equation dy/dx=xy with y(2)=4
A. √x2+6
B. -√x2+12
C. √x2+12
D. -√x2+6
E. x+√12
Short Answer
Consider the curve given by x2+3y2=1+3xy.
1) Show that dy/dx=3y-2x/6y-3x
Consider the differential equation dy/dx=x2(2y+1)
2) Find the particular solution y=f(x) to the given differential equation with the initial condition
f(0)=5.
Answer Key
Multiple Choice:
Choice C
Short Answer
1) 2x+6ydy/dx=0+3xdydx+3y
6ydy/dx-3xdy/dx=3y-2x
dy/dx(6y-3x)=3y-2x
dy/dx=37-2x/6y-3x
2) dy/dx=x2(2y+1)
dy=x2(2y+1)dx
(1/(2y+1))dy=x2dx
ln(2y+1)=⅓x3+C
ln(2(5)+1)= ⅓(0)3+C
ln(11)=C
ln(2y+1)= ⅓x3+ln(11)
2y+1=e^(x3/3)*11
2y=11e^(x3/3)-1
y=½(11e^(x3/3)-1)