Page 1
DIFFERENTIAL EQUATIONS
A differential equation is a mathematical equation that relates
some function with its derivatives
Order and Degree of a Differential Equation:
The order of the highest differential coefficient appearing in the
differential equation is called the order of the differential
equation, while the exponent of the highest differential
coefficient, when the differential equation is a polynomial in all
the differential coefficients, is known as the degree of the
differential equation.
Formation of Differential Equations:
Consider a family of curves f( x, y, ?
1
, ?
2
,…, ?
n
) = 0, where ?
1
,
?
2
,…, ?
n
are n independent parameters.
Solution of Differential Equation
First Order Differential Equations with Separable Variables
Let
? ?
? ?
?
Mx
dy
dx Ny
? N(y) dy = M(x) dx
Integrating both sides of (2), we get the solution viz.
? ? ? ? ??
??
N y dy M x dx c
Differential Equations Reducible to the Separable Variable
Type:
A differential equation of the form ?
dy
dx
f (ax + by + c) is solved by
writing ax + by + c = t.
Page 2
DIFFERENTIAL EQUATIONS
A differential equation is a mathematical equation that relates
some function with its derivatives
Order and Degree of a Differential Equation:
The order of the highest differential coefficient appearing in the
differential equation is called the order of the differential
equation, while the exponent of the highest differential
coefficient, when the differential equation is a polynomial in all
the differential coefficients, is known as the degree of the
differential equation.
Formation of Differential Equations:
Consider a family of curves f( x, y, ?
1
, ?
2
,…, ?
n
) = 0, where ?
1
,
?
2
,…, ?
n
are n independent parameters.
Solution of Differential Equation
First Order Differential Equations with Separable Variables
Let
? ?
? ?
?
Mx
dy
dx Ny
? N(y) dy = M(x) dx
Integrating both sides of (2), we get the solution viz.
? ? ? ? ??
??
N y dy M x dx c
Differential Equations Reducible to the Separable Variable
Type:
A differential equation of the form ?
dy
dx
f (ax + by + c) is solved by
writing ax + by + c = t.
Homogeneous Differential Equations:
Suppose we have a differential equation of the form ? ? ?
dy
f x,y ,
dx
where f(x,y) is a function of x and y and is of the form
?? ??
?? ??
?? ??
y x
F or F .
xy
These equations are solved by putting y = vx, where v ?v (x), a
function of x.
Equations Reducible to the Homogenous Form:
Equations of the form
??
?
??
dy ax by c
dx Ax By C
(aB ? Ab) can be reduced to a homogenous form by changing the
variables x, y to X, Y by writing x = X + h and y = Y + k;
where h, k are constants to be chosen so as to make the given
equation homogenous. We have
? ?
? ?
?
??
?
d Y k
dy dY
dx dX d X h
. Hence the given equation becomes
? ?
? ?
?
? ? ? ?
aX + bY + ah + bk + c
dY
dX AX BY Ah Bk C
Let h and k be so chosen as to satisfy the relation ah + bk + c = 0
and Ah + Bk + C =0.
These give
??
??
??
bC Bc Ac aC
hk
aB Ab aB Ab
which are meaningful except when aB = Ab.
?
?
?
dY aX bY
dX AX BY
can now be solved by means of the substitution Y = VX
In case aB = Ab, we write ax + by = t.
First Order Linear Differential Equations:
The most general form of this category of differential equations
is ??
dy
Py Q,
dx
, where P and Q are functions of x alone.
Page 3
DIFFERENTIAL EQUATIONS
A differential equation is a mathematical equation that relates
some function with its derivatives
Order and Degree of a Differential Equation:
The order of the highest differential coefficient appearing in the
differential equation is called the order of the differential
equation, while the exponent of the highest differential
coefficient, when the differential equation is a polynomial in all
the differential coefficients, is known as the degree of the
differential equation.
Formation of Differential Equations:
Consider a family of curves f( x, y, ?
1
, ?
2
,…, ?
n
) = 0, where ?
1
,
?
2
,…, ?
n
are n independent parameters.
Solution of Differential Equation
First Order Differential Equations with Separable Variables
Let
? ?
? ?
?
Mx
dy
dx Ny
? N(y) dy = M(x) dx
Integrating both sides of (2), we get the solution viz.
? ? ? ? ??
??
N y dy M x dx c
Differential Equations Reducible to the Separable Variable
Type:
A differential equation of the form ?
dy
dx
f (ax + by + c) is solved by
writing ax + by + c = t.
Homogeneous Differential Equations:
Suppose we have a differential equation of the form ? ? ?
dy
f x,y ,
dx
where f(x,y) is a function of x and y and is of the form
?? ??
?? ??
?? ??
y x
F or F .
xy
These equations are solved by putting y = vx, where v ?v (x), a
function of x.
Equations Reducible to the Homogenous Form:
Equations of the form
??
?
??
dy ax by c
dx Ax By C
(aB ? Ab) can be reduced to a homogenous form by changing the
variables x, y to X, Y by writing x = X + h and y = Y + k;
where h, k are constants to be chosen so as to make the given
equation homogenous. We have
? ?
? ?
?
??
?
d Y k
dy dY
dx dX d X h
. Hence the given equation becomes
? ?
? ?
?
? ? ? ?
aX + bY + ah + bk + c
dY
dX AX BY Ah Bk C
Let h and k be so chosen as to satisfy the relation ah + bk + c = 0
and Ah + Bk + C =0.
These give
??
??
??
bC Bc Ac aC
hk
aB Ab aB Ab
which are meaningful except when aB = Ab.
?
?
?
dY aX bY
dX AX BY
can now be solved by means of the substitution Y = VX
In case aB = Ab, we write ax + by = t.
First Order Linear Differential Equations:
The most general form of this category of differential equations
is ??
dy
Py Q,
dx
, where P and Q are functions of x alone.
We use here an Integrating factor, namely
?
pdx
e and the solution is
obtained by
??
??
?
pdx pdx
y e Q e dx c , where c is an arbitrary constant.
General Form of Variable Separation
(i) d(x + y) = dx + dy (ii) d(xy) = y dx + x dy
(iii) d
??
??
??
x
y
=
2
y dx-x dy
y
(iv) d
??
??
??
y
x
=
2
x dy-y dx
x
(v) d(log xy) =
ydx+xdy
xy
(vi) d
? ? ?
??
?
??
??
xdy ydx
y
log
x xy
(vii)
??
?
??
? ??
22
x+y x dy-y dx 1
d log
2 x-y
xy
(viii)
?
??
?
??
? ??
1
22
y x dy-y dx
d tan
x
xy
(ix)
? ?
?
??
??
?
1n
d f x,y
1n
=
? ?
? ? ? ?
?
n
f x,y
f x,y
(x) d
? ?
22
22
xdx ydx
xy
xy
?
??
?
ORTHOGONAL TRAJECTORY
Any curve, which cuts every member of a given family of curves
at right angle, is called an orthogonal trajectory of the family. For
example, each straight line passing through the origin, i.e. y = kx
is an orthogonal trajectory of the family of the circles x
2
+ y
2
= a
2
APPLICATION OF DIFFERENTIAL EQUATIONS
Geometrical Applications
We also use differential equations for finding the family of curves
for which some conditions involving the derivatives are given.
For this we proceed in the following way:
Equation of the tangent at a point (x, y) to the curve y = f(x) is
given by
Page 4
DIFFERENTIAL EQUATIONS
A differential equation is a mathematical equation that relates
some function with its derivatives
Order and Degree of a Differential Equation:
The order of the highest differential coefficient appearing in the
differential equation is called the order of the differential
equation, while the exponent of the highest differential
coefficient, when the differential equation is a polynomial in all
the differential coefficients, is known as the degree of the
differential equation.
Formation of Differential Equations:
Consider a family of curves f( x, y, ?
1
, ?
2
,…, ?
n
) = 0, where ?
1
,
?
2
,…, ?
n
are n independent parameters.
Solution of Differential Equation
First Order Differential Equations with Separable Variables
Let
? ?
? ?
?
Mx
dy
dx Ny
? N(y) dy = M(x) dx
Integrating both sides of (2), we get the solution viz.
? ? ? ? ??
??
N y dy M x dx c
Differential Equations Reducible to the Separable Variable
Type:
A differential equation of the form ?
dy
dx
f (ax + by + c) is solved by
writing ax + by + c = t.
Homogeneous Differential Equations:
Suppose we have a differential equation of the form ? ? ?
dy
f x,y ,
dx
where f(x,y) is a function of x and y and is of the form
?? ??
?? ??
?? ??
y x
F or F .
xy
These equations are solved by putting y = vx, where v ?v (x), a
function of x.
Equations Reducible to the Homogenous Form:
Equations of the form
??
?
??
dy ax by c
dx Ax By C
(aB ? Ab) can be reduced to a homogenous form by changing the
variables x, y to X, Y by writing x = X + h and y = Y + k;
where h, k are constants to be chosen so as to make the given
equation homogenous. We have
? ?
? ?
?
??
?
d Y k
dy dY
dx dX d X h
. Hence the given equation becomes
? ?
? ?
?
? ? ? ?
aX + bY + ah + bk + c
dY
dX AX BY Ah Bk C
Let h and k be so chosen as to satisfy the relation ah + bk + c = 0
and Ah + Bk + C =0.
These give
??
??
??
bC Bc Ac aC
hk
aB Ab aB Ab
which are meaningful except when aB = Ab.
?
?
?
dY aX bY
dX AX BY
can now be solved by means of the substitution Y = VX
In case aB = Ab, we write ax + by = t.
First Order Linear Differential Equations:
The most general form of this category of differential equations
is ??
dy
Py Q,
dx
, where P and Q are functions of x alone.
We use here an Integrating factor, namely
?
pdx
e and the solution is
obtained by
??
??
?
pdx pdx
y e Q e dx c , where c is an arbitrary constant.
General Form of Variable Separation
(i) d(x + y) = dx + dy (ii) d(xy) = y dx + x dy
(iii) d
??
??
??
x
y
=
2
y dx-x dy
y
(iv) d
??
??
??
y
x
=
2
x dy-y dx
x
(v) d(log xy) =
ydx+xdy
xy
(vi) d
? ? ?
??
?
??
??
xdy ydx
y
log
x xy
(vii)
??
?
??
? ??
22
x+y x dy-y dx 1
d log
2 x-y
xy
(viii)
?
??
?
??
? ??
1
22
y x dy-y dx
d tan
x
xy
(ix)
? ?
?
??
??
?
1n
d f x,y
1n
=
? ?
? ? ? ?
?
n
f x,y
f x,y
(x) d
? ?
22
22
xdx ydx
xy
xy
?
??
?
ORTHOGONAL TRAJECTORY
Any curve, which cuts every member of a given family of curves
at right angle, is called an orthogonal trajectory of the family. For
example, each straight line passing through the origin, i.e. y = kx
is an orthogonal trajectory of the family of the circles x
2
+ y
2
= a
2
APPLICATION OF DIFFERENTIAL EQUATIONS
Geometrical Applications
We also use differential equations for finding the family of curves
for which some conditions involving the derivatives are given.
For this we proceed in the following way:
Equation of the tangent at a point (x, y) to the curve y = f(x) is
given by
Y ?y = ? ? ?
dy
Xx
dx
, At the X axis, Y = 0, and X = x - y/
dy
dx
.
At the Y axis, X = 0, and Y = y ? x
dy
dx
.
Similar information can be obtained for normals by writing its
equation as (y - y)
dy
dx
+ (X-x) =0.
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