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Differential Equations
Any equation containing a derivative is called a differential equation.
Differential Equations
A function which satisfies the equation is called a solution to the differential equation.
Differential Equations
The general solution is a family of curves. Each member is a particular solution
.
Differential Equations
The order of the differential equation is the order of the highest derivative involved.
These are examples of FIRST ORDER differential equations
Differential Equations
The order of the differential equation is the order of the highest derivative involved.
These are examples of SECOND ORDER differential equations
3 Types of Differential Equations
Example:
3 Types of Differential Equations
Example:
3 Types of Differential Equations
Example:
3 Types of Differential Equations
Example:
This is the general solution
The constant can be written as part of the logarithm
Example:
Particular solution
Solve:
If when
Particular solution
Solve:
If when
Particular solution
Solve:
If when
Particular solution
Solve:
2nd Types of Differential Equations
)(yfdx
dy form the of is equation aldifferenti a When
dxdyyf
)(
1We can then integrate both sides.
dxyf
dy
)(This will obtain the general solution.
Example
A curve passes through the point (0, π/3) and its gradient at the point (x, y) is given by
Find the equation of the curve. (Sidebotham)
General Solution
A curve passes through the point (0, π/3) and its gradient at the point (x, y) is given by
Find the equation of the curve. (Sidebotham)
General Solution
A curve passes through the point (0, π/3) and its gradient at the point (x, y) is given by
Find the equation of the curve. (Sidebotham)
When the equation is of the form ( ) ( ), thendy
f x g ydx
1( )
( )dy f x dx
g y
( )( )dy
f x dxg y
Type 3
Solve the differential equation 2 3dy
xy xdx
Solve the differential equation 2 3dy
xy xdx
Solve the differential equation 2 3dy
xy xdx
Testing solutions
• Prove that is a solution to the differential equation
•
Do not try to integrate
•
Do not try to integrate
• Substitute