Summary

Videos

References

Method of solution to solve first order, first degree differential equations.

Consider the differential equation dy/dx = F(x,y)

If F(x, y) can be split into two functions, then the variable separable method is applied.

Ex:

Let dy/dx = (1+y^{2})/ ((1+x)xy)

= 1/x(1+x) . (1+y^{2})/ y

Here the first function is a function of x, and the other one is a function of y.

The right hand side is expressed as functions of x and y, so that the variables separable method can be applied.

â‡’ y/(1+y^{2}) dy/dx = 1/x(1+x)

â‡’ y/(1+y^{2}) dy = 1/x(1+x) dx

â‡’ y/(1+y^{2}) dy = (1/x - 1/(1+x)) dx (Integrating both the sides )

â‡’ âˆ« y/(1+y^{2}) dy = âˆ« (1/x - 1/(1+x)) dx

â‡’ 1/2 âˆ« 2y/(1+y^{2}) dy = âˆ« (1/x - 1/(1+x)) dx (Using formula âˆ« f'(x)/f(x) dx = log f(x) + C)

â‡’ 1/2 log(1+y^{2}) = log x - log(1+x) + log C

â‡’ log(1+y^{2})^{1/2} = log x - log(1+x) + log C

â‡’ log(1+y^{2})^{1/2} = log Cx/(1+x)

â‡’ (1+y^{2})^{1/2} = Cx/(1+x)

â‡’ (1+y^{2}) = (Cx/(1+x))^{2}

â‡’ (1+y^{2})(1+x)^{2} = Cx^{2}

Which is the solution of the given differential equation.

Method of solution to solve first order, first degree differential equations.

Consider the differential equation dy/dx = F(x,y)

If F(x, y) can be split into two functions, then the variable separable method is applied.

Ex:

Let dy/dx = (1+y^{2})/ ((1+x)xy)

= 1/x(1+x) . (1+y^{2})/ y

Here the first function is a function of x, and the other one is a function of y.

The right hand side is expressed as functions of x and y, so that the variables separable method can be applied.

â‡’ y/(1+y^{2}) dy/dx = 1/x(1+x)

â‡’ y/(1+y^{2}) dy = 1/x(1+x) dx

â‡’ y/(1+y^{2}) dy = (1/x - 1/(1+x)) dx (Integrating both the sides )

â‡’ âˆ« y/(1+y^{2}) dy = âˆ« (1/x - 1/(1+x)) dx

â‡’ 1/2 âˆ« 2y/(1+y^{2}) dy = âˆ« (1/x - 1/(1+x)) dx (Using formula âˆ« f'(x)/f(x) dx = log f(x) + C)

â‡’ 1/2 log(1+y^{2}) = log x - log(1+x) + log C

â‡’ log(1+y^{2})^{1/2} = log x - log(1+x) + log C

â‡’ log(1+y^{2})^{1/2} = log Cx/(1+x)

â‡’ (1+y^{2})^{1/2} = Cx/(1+x)

â‡’ (1+y^{2}) = (Cx/(1+x))^{2}

â‡’ (1+y^{2})(1+x)^{2} = Cx^{2}

Which is the solution of the given differential equation.