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Track your progress across all skills in your objective. Mark your confidence level and identify areas to focus on.
Track your progress:
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📖 = included in formula booklet • 🚫 = not in formula booklet
Track your progress:
Don't know
Working on it
Confident
📖 = included in formula booklet • 🚫 = not in formula booklet
Track your progress across all skills in your objective. Mark your confidence level and identify areas to focus on.
Track your progress:
Don't know
Working on it
Confident
📖 = included in formula booklet • 🚫 = not in formula booklet
Track your progress:
Don't know
Working on it
Confident
📖 = included in formula booklet • 🚫 = not in formula booklet
Track your progress across all skills in your objective. Mark your confidence level and identify areas to focus on.
Track your progress:
Don't know
Working on it
Confident
📖 = included in formula booklet • 🚫 = not in formula booklet
Track your progress:
Don't know
Working on it
Confident
📖 = included in formula booklet • 🚫 = not in formula booklet
The easiest differential equations to solve are the ones in the form
as we can simply integrate:
When you have a differential equation in the form
you can bring all the y's to one side and all the x's to the other:
And then integrate:
The solutions to differential equations will usually contain a constant of integration +C. These are called general solutions.
Often, we are given an initial condition, ie the value of y for a specific x, which we can use to solve for C. The result is the particular solution.
For a differential equation in the form
Multiply both sides by integrating factor (often called μ):
and notice the product rule on the LHS.
Let y=vx, then v=xy and dxdy=v+x⋅dxdv.
Note: On IB exams you will be told to use the substitution y=vx.
When we have a differential equation of the form dxdy=f(x,y), Euler's Method enables us to estimate the value of y for a specific x by starting with a known point and taking small steps towards the x-value we are interested in.
It works like this:
Start at the known point.
Find the slope at the current point.
Take a horizontal step of size h
Take a vertical step of size h times the slope found in 2.
Repeat 2-5 until the desired x-value is reached.
This is useful because in the physical world, we often know the state in which a system starts, and we have equations that model how the system will change.
It is very common for these physical models not to have exact solutions, so we need to use numerical methods, of which Euler's Method is one example.
Euler's Method is a technique for approximating numerical solutions to differential equations by taking small steps in the x and y directions in accordance with the differential equation.
Mathematically Euler's Method works as follows:
Start at a known point (x0,y0)
Pick a step size h such that x0+nh=xfinal for some integer n.
Repeat the following steps for each n until the desired x-value is reached:
Find the slope dxdy=f(xn,yn)
Find the next x value xn+1=xn+h📖.
Find the next y-value yn+1=yn+h×f(xn,yn)📖
Track your progress across all skills in your objective. Mark your confidence level and identify areas to focus on.
Track your progress:
Don't know
Working on it
Confident
📖 = included in formula booklet • 🚫 = not in formula booklet
Track your progress:
Don't know
Working on it
Confident
📖 = included in formula booklet • 🚫 = not in formula booklet
The easiest differential equations to solve are the ones in the form
as we can simply integrate:
When you have a differential equation in the form
you can bring all the y's to one side and all the x's to the other:
And then integrate:
The solutions to differential equations will usually contain a constant of integration +C. These are called general solutions.
Often, we are given an initial condition, ie the value of y for a specific x, which we can use to solve for C. The result is the particular solution.
For a differential equation in the form
Multiply both sides by integrating factor (often called μ):
and notice the product rule on the LHS.
Let y=vx, then v=xy and dxdy=v+x⋅dxdv.
Note: On IB exams you will be told to use the substitution y=vx.
When we have a differential equation of the form dxdy=f(x,y), Euler's Method enables us to estimate the value of y for a specific x by starting with a known point and taking small steps towards the x-value we are interested in.
It works like this:
Start at the known point.
Find the slope at the current point.
Take a horizontal step of size h
Take a vertical step of size h times the slope found in 2.
Repeat 2-5 until the desired x-value is reached.
This is useful because in the physical world, we often know the state in which a system starts, and we have equations that model how the system will change.
It is very common for these physical models not to have exact solutions, so we need to use numerical methods, of which Euler's Method is one example.
Euler's Method is a technique for approximating numerical solutions to differential equations by taking small steps in the x and y directions in accordance with the differential equation.
Mathematically Euler's Method works as follows:
Start at a known point (x0,y0)
Pick a step size h such that x0+nh=xfinal for some integer n.
Repeat the following steps for each n until the desired x-value is reached:
Find the slope dxdy=f(xn,yn)
Find the next x value xn+1=xn+h📖.
Find the next y-value yn+1=yn+h×f(xn,yn)📖