IBHM 446 472 .pdf


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16 Integration 3 – Applications

16.3 Solving differential equations
by separating variables
Equations in which the variables are separable can be written in the form
dy
 f1x2. These can then be solved by integrating both sides with respect to x.
g1y2
dx

Method
dy
 f1x2
dx

1. Put in the form g1y2

This gives an equation in the form

冮 g1y2 dx dx  冮 f1x2 dx which simplifies to
dy

冮 g1y2 dy  冮 f1x2 dx. The variables are now separated onto opposite sides of the
equation.
2. Integrate both sides with respect to x.
3. Perform the integration.

Example
Find the general solution to the differential equation

dy
4x  1

.
dx
2y

Following step 1:
dy
2y  4x  1
dx
Following step 2 and step 3:
dy
2y dx  14x  12 dx
dx





冮 2y dy  冮 14x  12 dx

1
1

Since there is an integral on
each side of the equation, a
constant of integration is
theoretically needed on each
side. For simplicity, these are
usually combined and written
as one constant.

2y2
4x2
 k1 
 x  k2
2
2
˛

˛

˛

˛

1 y2  2x2  x  k
˛

˛

1 y  ;22x2  x  k
˛

Example
Solve to find the general solution of the equation ex

dy
x
 2
.
dx
y 1
˛

Following step 1:
dy
1y2  12
 xex
dx
Following step 2 and step 3:
dy
1y2  12 dx  xex dx
dx
˛




1 冮 1y  12 dy  冮 xe
˛

2

˛

452

x

dx


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