What is Differential equation?If y is a function of x, then we

denote it as y = f(x). Here x is called an independent variable and y is called a dependent variable.

If there is a equation dy/dx = g(x) ,then this equation contains the variable x and derivative of y w.r.t x. This type of an equation is known as a Differential Equation.

Order of Differential Equation

Order of the highest order derivative of the dependent variable with respect to the independent variable occurring in a given differential equation is called the order of differential equation.

E.g. – 1st order equation 2nd order equation

Degree of Differential EquationWhen a differential equation is in

a polynomial form in derivatives, the highest power of the highest order derivative occuring in the differential equation is called the degree of the differential equation.

E.g. – Degree – 1 ,(d²y/dx) + dy/dx = 0

Degree – 2 , (d²y/dx)² + dy/dx = 0

Solution of differential equations of the first order and first degreeDifferential equations of 1st order

can be solved by many methods ,some of the methods are as follows :-

1. Variable Separable Method 2. Exact equation method 3. Homogenous equation method 4. Linear equation method

Solution of differential equations of the first order and first degree 5. Non-Linear Equation method

(Bernoulli's equation) 6. Non-Exact Equation method

( , )y f x y

Linear Non-linear

Integrating Factor

Separable Homogeneous Exact

IntegratingFactor

Transform to ExactTransform to separable

Important Forms of the methodHere are some important forms

of the method through which we can know the form of equation and then use or apply the method which is required :-

1. Variable Separable method – Equation is in the form of :

dy/dx = M(x)/N(y) ordy/dx = M(x)N(y)

Important Forms of the method 2. Exact equation method – equation

is in the form of :Mdx + Ndy = 0 --- 1If , ∂M/∂y = ∂N/∂x Then the above equation 1 is Exact

equation 3. Homogenous equation method -

equation is in the form of :dy/dx = x²y + x³y + xy²/x³ - y³

(Example)

Important Forms of the method 4. Linear equation method -

equation is in the form of :Form -1 : dy/dx + Py = Q (x form)Form - 2 : dx/dy + Px = Q (y form) 5. Non-Linear Equation method

(Bernoulli's equation) - equation is in the form of :

Dy/dx – 2ytanx = y²tan²x (Example)

Important Forms of the method 6. Non-Exact Equation method -

equation is in the form of :Type – 1 : Mdx + Ndy = 0F(x) = 1/N (∂M/∂y - ∂N/∂x) (x form)Type – 2 : Mdx + Ndy = 0F(y) = 1/M (∂N/∂x - ∂M/∂y)

1st Order DE - Homogeneous EquationsHomogeneous Function

f (x,y) is called homogenous of degree n if : y,xfy,xf n Examples:

yxxy,xf 34 homogeneous of degree 4

yxfyxx

yxxyxf,

,4344

34

yxxyxf cos sin, 2 non-homogeneous

yxfyxxyxxyxf

n ,

cos sin

cos sin,22

2

1st Order DE - Homogeneous EquationsThe differential equation M(x,y)dx + N(x,y)dy = 0 is homogeneous if M(x,y) and N(x,y) are homogeneous and of the same degree

Solution :1. Use the transformation to : dvxdxvdyvxy

2. The equation become separable equation:

0,, dvvxQdxvxP

3. Use solution method for separable equation

Cdvvgvgdx

xfxf

1

2

2

1

4. After integrating, v is replaced by y/x

Variable – separable example 1. dy/dx = x.y =dy/y = xdx =∫dy/y = ∫xdx =logy = x²/2 + c

Exact Equation Example 1. xdy/dx + y + 1 = 0 =xdy + (y + 1)dx = 0 here , M = y + 1 , N = x ∂M/∂y = 1 , ∂N/∂x = 1 therefore , ∂M/∂y = ∂N/∂x here the given equation is an

exact equation ∫Mdx(y constant) + ∫(terms of N

not containing x)dy = c

Exact Equation Example =∫(y + 1)dx (y constant) + ∫0.dx

= c = x(y + 1) = c