Popular methods of solving differential equation of first order and first degree.

 Popular methods of solving differential equation of first order and first degree. 

1:- Variable separable method.

 2:- Reducible to Variable separable method. 

 3:- Homogeneous differential equation .

 4:- Reducible to Homogeneous differential equation .

5:- Linear differential equation.

6:- Reducible to Linear differential equation.

7:- Exact differential equation.

 8:- Reducible to Exact differential equation. 

1:- Variable separable method.

General Format;

f(x)dx + g(y)dy = 0

Solution is its integration, 

ie, ∫f(x)dx + ∫g(y)dy = 0

Example,

(1+x²)dy = (1+y²)dx

=》∫dy/(1+y²)=∫dx/(1+x²)

=》tan‐¹y = tan‐¹x +c 

Or

=》tan‐¹y = tan‐¹x +tan-¹c

=》tan‐¹y - tan‐¹x = tan-¹c

=》tan-¹[(y-x)/1+yx] = tan-¹c

=》[(y-x)/1+yx] = c

=》(y-x) = c(1+yx)

is the solution of this differential equation.

REDUCIBLE TO VARIABLE SEPARATION METHOD

When x and y can't be separated ,

Example:-   (x+y)²dy/dx = a²

Then we let Linear form of ( ax+by+c) = v

Like ,

Let , x+y=v .................eq(1)

differenting with respect to x

=》 1 + dy/dx = dv/dx

=》 dy/dx = dv/dx - 1

by putting the value of dy/dx in question, so that,

=》(x+y)² (dv/dx - 1)= a²

Putting x+y as v , from eq(1)

=》(v)² (dv/dx - 1)= a²

=》dv/dx = a²/v² +1

=》dv/dx = (a²+v²)/v²

=》v²dv/ (a²+v²)= dx

Solution is, 

=》∫v²dv/ (a²+v²)= ∫dx

=》∫{1-a²/(a²+v²)} dv= ∫dx

=》v - a²(1/a tan-¹x/a) = x + c

=》v - a.tan-¹(x/a) = x + c

replacing the value of v from eq(1) x+y=v

=》x+y - a.tan-¹(x/a) = x + c

=》y - a.tan-¹(x/a) = c

 is the final answer. 🙌 

Another example 

Example number

 2:- dy/dx =(3y+x+6)²

of Reducible to Variable separable method. 

Solve,

Assuming the linear form 3y+x+6=v.............eq(1)

differenting with respect to x

=》3dy/dx +1 = dv/dx

=》dy/dx = (dv/dx -1)1/3

by putting the value of dy/dx in question, so that,

 =》(dv/dx -1)1/3 =(3y+x+6)²

Putting (3y+x+6) as v , from eq(1)

 =》(dv/dx -1)1/3 =(v)²

 =》(dv/dx -1) =3(v)²

 =》dv/dx = 3(v)²+1

=》dv / (3v)²+1) = dx

Solution is its integration, 

=》∫dv / (3v)²+1) = ∫dx

After integration replace the value of v and get the real answer.