Different forms of first order and first degree differential equations

(a) Variable separable differential equations

f (x) dx = g(y)dy

Method Integrate it both sides i.e.

(b) Reducible to variable separable differential equations

Some times directly differential equation does not take form of the type f (x) dx = g(y)dy, but after some substitution we get this form. For example

= x + y

Put x + y = t ⇒ 1 + =

So − 1 = t ⇒ =dx

(Now this reduces to variable separable differential equation)

(c) Homogeneous Differential Equation

We are familiar with homogeneous equations in x and y (where degree of each term in x and y is same). Now homogeneous differential equation is where degree of each term is same (here we consider degree of any derivative as zero). For example

x² + y².+ xy = 0 is a homogeneous differential equation

Method = f(y/x)

Now put  y/x = t ⇒ y = t x

or = t + x.

So t + x. = f (t)

⇒

This reduces to variable separable differential equation.

(d) Reducible to homogeneous differential equation

=

Put x = X + h and y = Y + k (h, k are constants)

=

⇒ =

put a₁h + b₁k + c₁ = 0

a₂h + b₂k + c₂ = 0

Solve for h and k to find h and k.

Now =

So this reduces to homogeneous differential equation.

And if , then put a1x + b1y = t and then it reduces to variable separable, differential equation.

(e) Linear Differential Equations

Differential equations of the form + Py = Q (where P and Q are functions of x) is linear differential equation.

Method Multiply it with R (a function of x)

R. + R.Py = R.Q.

Now Let R + R.Py = (R.y)

R+ R.Py = R. + y.

⇒ y.= R.Py

⇒ = P.dx

⇒ R = (We call it Integrating factor and denote it by I.F.)

Now (R.y) = R.Q

⇒ R.y = dx

(Where R is integrating factor).

(f) Reducible To Linear Differential Equation

T(y) + P S(y) = Q (Where P and Q are functions of x)

and = T(y)

Method Put S(y) = z, then

+ P.z = Q