Hg
Name: Class 12 STICK TO YOUR WALL IN STUDY AREA
___________________________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________________________
The Hg Classes (8
th
to 12
th
) By: Er Hershit Goyal (B.Tech. IIT BHU), 134-SF, Woodstock Floors, Nirvana Country, Sector 50, GURUGRAM +91 9599697178.
fb.me/thehgclasses linkedin.com/company/the-hg-classes instagram.com/the_hg_classes g.page/the-hg-classes-gurugram thehgclasses.co.in
DIFFERENTIAL EQUATIONS
• Definition: An equation involving derivative(s) of the dependent variable w.r.t the independent variable (or
variables) is known as a differential equation.
• If derivatives are w.r.t only 1 independent variable, then it is called an ordinary differential equation.
• If differential equation contains derivatives w.r.t more than 1 independent variable, then it is called a partial
differential equation.
• Order of a differential equation is the order of the highest ordered derivative occurring in it.
• Degree of a differential equation is only defined for a polynomial equation in derivatives. It is the highest power
of the highest order derivative in it.
• Both order and degree (if defined) are always positive integers.
Examples:
Differential Equation
Order
Degree
dy/dx + 3x = 0
1
1
(dy/dx)
2
+ 5(dy/dx) = 0
1
2
d
2
y/dx
2
+ dy/dx + 9x = 0
2
1
d
2
y/dx
2
+ dy/dx + 9 sin x = 0
2
1
x dy/dx + y = 0
1
1
dy/dx + sin (dy/dx) = 0
1
not defined (as not a polynomial equation)
y’’’ + y
2
+ e
y
= 0
3
1
• Solution of the differential equation:
The function which satisfies the differential equation is called its solution.
o General solution: Solution contain as many arbitrary constants as the order of differential equation. This
represents a family of curves.
o Particular solution: Solution free from any arbitrary constant.
• Formation of a differential equation whose general solution is given
o Step 1: Differentiate the given solution function n number of times where n = no. of arbitrary constants
o Step 2: Eliminate the arbitrary constants from the n+1 equations obtained above including the original
function.
Thus, Order of a differential equation representing a family of curves = number of arbitrary constants present in
the equation corresponding to the family of curves.
• Variable Separable Method of solving a differential equation (order 1, degree 1)
o Step 1: Separate the variables i.e. Bring terms contains x and y on either side of the equal sign.
o Step 2: Integrate both sides.
• Homogeneous differential equation
o Homogeneous function: A function F(x, y) is a homogeneous fn of degree n,
if F(kx, ky) = k
n
F(x, y), for any non zero constant k.
In other words, F(x, y) can be expressed as either x
n
g(y/x) or y
n
h(x/y).
o An equation dy/dx = F(x, y) is said to be homogeneous, if F(x, y) is homogeneous function of degree 0.
o A homogeneous differential equation dy/dx = g(y/x) can be solved by the substitution y = vx.
o A homogeneous differential equation dx/dy = h(x/y) can be solved by the substitution x = vy.
Hg
Name: Class 12 STICK TO YOUR WALL IN STUDY AREA
___________________________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________________________
The Hg Classes (8
th
to 12
th
) By: Er Hershit Goyal (B.Tech. IIT BHU), 134-SF, Woodstock Floors, Nirvana Country, Sector 50, GURUGRAM +91 9599697178.
fb.me/thehgclasses linkedin.com/company/the-hg-classes instagram.com/the_hg_classes g.page/the-hg-classes-gurugram thehgclasses.co.in
• Linear differential equation
o An equation of the form dy/dx + Py = Q, where P and Q are either constants or functions of x only, is
known as a first order Linear differential equation.
o Steps to solve a first order linear differential equation:
▪ Write it in the form dy/dx + Py = Q
▪ Find the I.F (integrating factor) =
▪ The solution is given by y * I.F =
dx + C
o Similarly, an equation of the form dx/dy + Px = Q, where P and Q are either constants of functions of y
only is also a Linear differential equation.
▪ The solution is given by x * I.F =
+ C, where I.F =
