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Calculus study Notes

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Calc 2 Lecture Notes Section 7.2 Page 1 of 8
Section 7.2: Separable Differential Equations
Big idea: There are many different techniques for solving diff. eqs. (Not DIFFY “Q”).
Differential equations get grouped according to their method of solution. Differential equations
that can be solved by isolating the function and the independent variable on opposite sides of the
equation are called “SEPARABLE”.
Separable differential equations have the property:
( ) ( ) ( )y x u x v y
= ×
Thus their solution is given by the formula:
1
( ) ( )
( )
1
( )
( )
y x u x
v y
dy u x dx c
v y
=
= +
Big skill:. To be able to recognize separable differential equations, and to solve them by
separating the “variables”.
Example:
( )
( )
( )
( )
( )
2 2
2
2 1
2 1
2 1
2 1
2 1
1
2
2 2
x
y x
y x
yy x
dy
y x
dx
ydy x dx
ydy x dx
y x x c
y x x x c
+
=
= +
= +
= +
= +
= + +
= + +

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Calc 2 Lecture Notes Section 7.2 Page 2 of 8
Another (Hard) Example:
( ) ( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
( ) ( ) ( )
2
2
2
2
2
2 2
2
2 cos 1
2
cos 1
1
2
cos 1
1
2
cos 1
1
2
cos 1
cos 1
1
cos 1
cos 1
cos 1
cos 1
cos 1
sin
cot csc csc
csc( ) cot( )
y x x y x
y
x
y
dy
x
dx
y
dy xdx
y
dy xdx
y
y
dy x c
y
y
y
dy x c
y
y
dy x c
y
y y dy y dy x c
y y x c
=
=
× =
× =
× =
+
× = +
+
+
= +
+
= +
= +
+ = +
Notice: the solution can only be stated implicitly.

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Calc 2 Lecture Notes Section 7.2 Page 3 of 8
Another Example:
( ) ( )
( )
( )
( )
( )
( )
2
2
2
2 1
2
1
1
2
1
1
2
1
1
2
1
ln 1
1 ^ ( )
( ) 1 ^ ( )
y x x y x
y
x
y
dy
x
y dx
dy xdx
y
dy xdx
y
y x c
y e x c
y x e x c
=
=
× =
× =
× =
= +
= +
= + +
Family of solutions: all the different graphs for all the c values…

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Section 7.2: Separable Differential EquationsBig idea: There are many different techniques for solving diff. eqs. (Not DIFFY "Q"). Differential equations get grouped according to their method of solution. Differential equations that can be solved by isolating the function and the independent variable on opposite sides of the equation are called "SEPARABLE".Separable differential equations have the property: Thus their solution is given by the formula: Big skill:. To be able to recognize separable differential equations, and to solve them by separating the "variables".Example:Another (Hard) Example:Notice: the solution can only be stated implicitly.Another Example:Family of solutions: all the different graphs for all the c values?Initial value problem (IVP) ? initial conditions: How we pick a given value for c?We can only solve for the constant of integration c if we are given an initial condition (i.e. a point on a curve).For example: IF y(0) = 1.5THEN 1+e^(0+c) = 1.5 ? e^c = 0.5 ? c = ln(0.5) ? y(x) = 1 + e^(x2 + ln(0.5))Hint: It is often easier to solve for c before you get the final form of y(x).Another IVP example:If the IVP is y(2) = ?/4, then Logistic growth: A model that reflects the fact that as a population grows exponentially, it is also filling up its environment, which limits the rate of growth. If k is the growth rate, and M is the maximum carrying capacity of the environment, then the mathematical m ...
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