Chapter 9
DIFFERENTIAL EQUATIONS
9.1 Overview
(i) An equation involving derivative (derivatives) of the dependent variable with
respect to independent variable (variables) is called a differential equation.
(ii) A differential equation involving derivatives of the dependent variable with
respect to only one independent variable is called an ordinary differential
equation and a differential equation involving derivatives with respect to more
than one independent variables is called a partial differential equation.
(iii) Order of a differential equation is the order of the highest order derivative
occurring in the differential equation.
(iv) Degree of a differential equation is defined if it is a polynomial equation in its
derivatives.
(v) Degree (when defined) of a differential equation is the highest power (positive
integer only) of the highest order derivative in it.
(vi) A relation between involved variables, which satisfy the given differential
equation is called its solution. The solution which contains as many arbitrary
constants as the order of the differential equation is called the general solution
and the solution free from arbitrary constants is called particular solution.
(vii) To form a differential equation from a given function, we differentiate the
function successively as many times as the number of arbitrary constants in the
given function and then eliminate the arbitrary constants.
(viii) The order of a differential equation representing a family of curves is same as
the number of arbitrary constants present in the equation corresponding to the
family of curves.
(ix) ‘Variable separable method’ is used to solve such an equation in which variables
can be separated completely, i.e., terms containing x should remain with dx and
terms containing y should remain with dy.
, 180 MATHEMATICS
(x) A function F (x, y) is said to be a homogeneous function of degree n if
F (λx, λy )= λn F (x, y) for some non-zero constant λ.
dy
(xi) A differential equation which can be expressed in the form = F (x, y) or
dx
dx
= G (x, y), where F (x, y) and G (x, y) are homogeneous functions of degree
dy
zero, is called a homogeneous differential equation.
dy
(xii) To solve a homogeneous differential equation of the type = F (x, y), we make
dx
substitution y = vx and to solve a homogeneous differential equation of the type
dx
= G (x, y), we make substitution x = vy.
dy
dy
(xiii) A differential equation of the form + Py = Q, where P and Q are constants or
dx
functions of x only is known as a first order linear differential equation. Solution
of such a differential equation is given by y (I.F.) = ∫ ( Q × I.F.) dx + C, where
I.F. (Integrating Factor) = e ∫ .
Pdx
dx
(xiv) Another form of first order linear differential equation is + P1x = Q1, where
dy
P1 and Q1 are constants or functions of y only. Solution of such a differential
equation is given by x (I.F.) = ∫ ( Q1 × I.F.) dy + C, where I.F. = e∫ P dy .
1
9.2 Solved Examples
Short Answer (S.A.)
Example 1 Find the differential equation of the family of curves y = Ae2x + B.e–2x.
Solution y = Ae2x + B.e–2x
, DIFFERENTIAL EQUATIONS 181
dy d2y
= 2Ae2x – 2 B.e–2x and = 4Ae2x + 4Be–2x
dx dx 2
d2y d2y
Thus = 4y i.e., 2 – 4y = 0.
dx 2 dx
dy y
Example 2 Find the general solution of the differential equation = .
dx x
dy y dy dx dy dx
Solution =
dx x
⇒
y
=
x
⇒ ∫ y
= ∫ x
⇒ logy = logx + logc ⇒ y = cx
dy
Example 3 Given that = yex and x = 0, y = e. Find the value of y when x = 1.
dx
dy dy
Solution
dx
= yex ⇒ ∫ y
= ∫ e dx
x
⇒ logy = ex + c
Substituting x = 0 and y = e,we get loge = e0 + c, i.e., c = 0 ( loge = 1)
Therefore, log y = ex.
Now, substituting x = 1 in the above, we get log y = e ⇒ y = ee.
dy y
Example 4 Solve the differential equation + = x2.
dx x
dy
Solution The equation is of the type + Py = Q , which is a linear differential
dx
equation.
1
Now I.F. = ∫ x dx = e logx
= x.
Therefore, solution of the given differential equation is
DIFFERENTIAL EQUATIONS
9.1 Overview
(i) An equation involving derivative (derivatives) of the dependent variable with
respect to independent variable (variables) is called a differential equation.
(ii) A differential equation involving derivatives of the dependent variable with
respect to only one independent variable is called an ordinary differential
equation and a differential equation involving derivatives with respect to more
than one independent variables is called a partial differential equation.
(iii) Order of a differential equation is the order of the highest order derivative
occurring in the differential equation.
(iv) Degree of a differential equation is defined if it is a polynomial equation in its
derivatives.
(v) Degree (when defined) of a differential equation is the highest power (positive
integer only) of the highest order derivative in it.
(vi) A relation between involved variables, which satisfy the given differential
equation is called its solution. The solution which contains as many arbitrary
constants as the order of the differential equation is called the general solution
and the solution free from arbitrary constants is called particular solution.
(vii) To form a differential equation from a given function, we differentiate the
function successively as many times as the number of arbitrary constants in the
given function and then eliminate the arbitrary constants.
(viii) The order of a differential equation representing a family of curves is same as
the number of arbitrary constants present in the equation corresponding to the
family of curves.
(ix) ‘Variable separable method’ is used to solve such an equation in which variables
can be separated completely, i.e., terms containing x should remain with dx and
terms containing y should remain with dy.
, 180 MATHEMATICS
(x) A function F (x, y) is said to be a homogeneous function of degree n if
F (λx, λy )= λn F (x, y) for some non-zero constant λ.
dy
(xi) A differential equation which can be expressed in the form = F (x, y) or
dx
dx
= G (x, y), where F (x, y) and G (x, y) are homogeneous functions of degree
dy
zero, is called a homogeneous differential equation.
dy
(xii) To solve a homogeneous differential equation of the type = F (x, y), we make
dx
substitution y = vx and to solve a homogeneous differential equation of the type
dx
= G (x, y), we make substitution x = vy.
dy
dy
(xiii) A differential equation of the form + Py = Q, where P and Q are constants or
dx
functions of x only is known as a first order linear differential equation. Solution
of such a differential equation is given by y (I.F.) = ∫ ( Q × I.F.) dx + C, where
I.F. (Integrating Factor) = e ∫ .
Pdx
dx
(xiv) Another form of first order linear differential equation is + P1x = Q1, where
dy
P1 and Q1 are constants or functions of y only. Solution of such a differential
equation is given by x (I.F.) = ∫ ( Q1 × I.F.) dy + C, where I.F. = e∫ P dy .
1
9.2 Solved Examples
Short Answer (S.A.)
Example 1 Find the differential equation of the family of curves y = Ae2x + B.e–2x.
Solution y = Ae2x + B.e–2x
, DIFFERENTIAL EQUATIONS 181
dy d2y
= 2Ae2x – 2 B.e–2x and = 4Ae2x + 4Be–2x
dx dx 2
d2y d2y
Thus = 4y i.e., 2 – 4y = 0.
dx 2 dx
dy y
Example 2 Find the general solution of the differential equation = .
dx x
dy y dy dx dy dx
Solution =
dx x
⇒
y
=
x
⇒ ∫ y
= ∫ x
⇒ logy = logx + logc ⇒ y = cx
dy
Example 3 Given that = yex and x = 0, y = e. Find the value of y when x = 1.
dx
dy dy
Solution
dx
= yex ⇒ ∫ y
= ∫ e dx
x
⇒ logy = ex + c
Substituting x = 0 and y = e,we get loge = e0 + c, i.e., c = 0 ( loge = 1)
Therefore, log y = ex.
Now, substituting x = 1 in the above, we get log y = e ⇒ y = ee.
dy y
Example 4 Solve the differential equation + = x2.
dx x
dy
Solution The equation is of the type + Py = Q , which is a linear differential
dx
equation.
1
Now I.F. = ∫ x dx = e logx
= x.
Therefore, solution of the given differential equation is
