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## AP®︎/College Calculus AB

### Course: AP®︎/College Calculus AB>Unit 7

Lesson 4: Reasoning using slope fields

# Approximating solution curves in slope fields

Given the slope field of a differential equation, we can sketch various solutions to the equation.

## Want to join the conversation?

• does slope field give the exact solution of differential equation?
if no then why do we draw slope field?
• You are essentially correct. The slope field is utilized when you want to see the tendencies of solutions to a DE, given that the solutions pass through a certain localized area or set of points. The beauty of slope field diagrams is that they can be drawn without actually solving the DE. This is a very useful tool often employed by people who study mathematical biology, or changes in populations over time (due to predation, climate, etc). Slope fields allow these people to view the probable trends of a certain population based on its conditioning factors without actually solving their DE's.
• is it possible to find solution if point is not given ?
• Yes, in differential equations you can find what is called a "general solution" even if no initial conditions are given. If you do have initial condition then you can transform your general solution into a "particular solution".

With the slope field visualisation, you saw Sal draw several particular solutions, each one dependent of the initial conditions that he choose.
• How can a value be a solution for a differential equation. Didn't Mr.Khan mentioned the solutions for differential equations are functions or class or functions?
• Absolutely correct but it could also be a function that is not dependent on x. For example, the differential of y=3x+2 is simply y'=3, and so the value 3 is a solution for the differential equation
• sal took x=1 and y=6 so slope=-2 ? i am curios what does the slope=1.5342 or slope=pi,or golden ratio look ? sorry for my bad english , hope you got the point
• As the differential equation dy/dx is a function of y, plugging in the y-value 6 gives
dy/dx = 6/6 * (4-6) = 1 *-2 = -2,
the slope you mentioned. If you look at the point (1, 6) on the slope field diagram, you can see a short downward sloping line, of approximately slope -2.
If the slope were pi at a point, you would see an upward sloping line of approximately 3.14159... . We can solve for which points this would be at:
dy/dx = pi = y/6 * (4-y)
pi*6 = 4y-y^2 //multiply both sides by 6, then distribute on the right
0 = -1*y^2 +4* y - 6*pi //add 6*pi to the right side
If you solve the quadratic, you should get the y-values where the slope is pi. You can do this similarly for any slope value you wish.

I hope this helps!
• Can we get a solution in which for one x we get two or more y values . I mean not exactly function we normally see but a multivalued function.
• well, actyally the dy/dx is the slope of the tiny straights we design? for example for x=1 y=1 dy/dx=1/2 we design a straight with angle 1/2 degrees??
(1 vote)
• Not the angle, but the slope. The slope of the segment you draw at (1,1) would be 1/2. The angle would be arctan(1/2) from the horizontal.
• how can you know that the solutions don't do something wacky between the slope segments you draw?

it seems intuitive enough here, but with more complicated equations, it seems like it could get messy
• That's the neat part: you don't.

Hence the title: "approximating" solution curves. You use the slope field to get a general idea of how the curve looks. But, how the curve actually looks will be found out by solving the DE and plotting the resultant curve equation.
• Wouldn't y=4 just be the critical point? Why would it be the solution?