Question

Solve the following Linear differential equation:

(2x – 10y³)dy + y dx = 0

Answer 1

The given differential equation may be written as

y dx = – (2x – 10y³)dy

`y  ("d"x)/("d"y) = - 2x + 10y^3`

`÷ y, y/y ("d"x)/("d"y) + 2/y x = (10y^3)/y`

`("d"x)/("d"y) + (2/y)x = 10y^2`

This is of the form `("d"x)/("d"y) + "P"x` = Q

Where P = `2/y`

Q = 10y²

Thus, the given equation is linear.

I.F = `"e"^(int"Pd"y)`

= `"e"^(int 2/y  "d"y)`

= `"e"^(2logy)`

= y²

So, the required solution is

x × I.F = `int ("Q" xx "I.F")  "d"y + "c"`

xy² = `int 10 y^2 xx y^2  "d"y + "c"`

- `int 10 y^4  "d"y + "c"`

= `(10y^5)/5 + "c"`

= 2y⁵ + c

xy² = 2y⁵ + c is a required solution.