Question

Solve the following Linear differential equation:

`("d"y)/("d"x) = (sin^2x)/(1 + x^3) - (3x^2)/(1 + x^3) y`

Answer 1

The equation can be written as

`("d"y)/("d"x) = (sin^2x)/(1 + x^3)y - (3x^2)/(1 + x^3)`

This equation is of the form

`("d"y)/("d"x) + "P"y` = Q

Where P = `(3x^2)/(1 + x^2)`

Q = `(sin^2x)/(1 + x^3)`

`int "Pd"x = int (3x^2)/(1 + x^2)  "d"x`

= `log(1 + x^3)`

∴ I.F = `"e"^(int "Pd"x)`

= `"e"^(log (1 +x^3)`

= `(1 + x^3)`

∴ The solution is `y  "e"^(int"Pd"x)`

= `int "Q"  "e"^(int "Pd"x)  "d"x + "c"`

`y(1 + x^3) = int (sin^2x)/(1 + x^3) xx (1 + x^3)  "d"x + "c"`

= `int sin^2x  "d"x + "c"`

= `int (1 - cos 2x)/2  "d"x + "c"`

`y(1 + x^3) = int(1/2 - (cos2x)/2) "d"x + "c"`

= `x/2 - (sin 2x)/(2 xx 2) + "c"`

`y(1 + x^3) = x/2 - (sin 2x)/4 + "c"` is a required solution.