Separable Differential Equations
Definition and Solution of a Separable Differential
Equation
A differential equation is called separable if it can be written as
Steps To Solve a Separable Differential Equation
To solve a separable differential equation

Get all the
y's
on the left hand side of the equation and all of
the x's on the right hand side.

Integrate both sides.

Plug in the given values to find the constant of integration
(C)

Solve for
y
Example:
Solve
dy/dx = y(3  x); y(0 )= 5

dy/y = (3  x) dx

ln y = 3x  x^{2} / 2 + C

ln 5 = 0 + 0 + C
C = ln 5

Exercises
Solve the following differential equations

dy/dx = x/y; y(0) = 1

dy/dx = x(x+1); y(1) = 1

2xy + dy/dx = x; y(0) = 2
Homogeneous Differential Equations
A differential equation is called a homogeneous
differential equation if it can be written in
the form
where
M and
N are of the same degree. To solve a homogeneous
differential equation follow the steps below:
Steps For Solving a Homogeneous Differential Equation

Rewrite the differential in
homogeneous form
M(x,y)dx + N(x,y)dy
= 0

Make the substitution y = vx
where v
is a variable.

Then use the product rule to get
dy = vdx + xdv

Substitute to rewrite the differential
equation in terms of v
and
x
only

Divide by x^{d}
where d is the degree of the polynomials M and
N.

Follow the steps for solving separable
differential equations.

Resubstitute v = y/x.
Example
Solve
y' = (x + y)/(x  y)
Solution

dy
x + y
=
dx
x  y
(x + y)dx + (y  x)dy
= 0

y = vx

dy = vdx + xdv

(x + vx)dx + (vx  x)(vdx + xdv) = 0

(1 + v)dx + (v  1)(vdx + xdv) = 0

[(1 + v) + (v^{2}  v)]dx +
(xv  x)dv = 0
[1 + v^{2}]dx
= (x  xv)dv
1
1  v
dx =
dv
x
1 + v^{2}
1
v
=
dv 
dv
1 + v^{2}
1 + v^{2}

At this point of the class, we do not
know the integral of 1/(1 + v^{2}). In a later section, we
will see that an antiderivative is arctan(v), hence
lnx = arctan(v)  1/2 ln(1 + v^{2}) + C

lnx = arctan(y/x)  1/2 ln(1 + (y/x)^{2})
+ C
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