# 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

dy

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

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
 xex
dx
Following step 2 and step 3:
dy
1y2  12 dx  xex dx
dx
˛

1 冮 1y  12 dy  冮 xe
˛

2

˛

452

x

dx

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