Question

Form the differential equation having for its general solution y = ax² + bx

Answer 1

y = ax² + bx  ......(1)

Since we have two arbitrary constants, differentiative twice.

`("d"y)/("d"x) = 2"a"x + "b"` ......(2)

`("d"^2y)/("d"x^2)` = 2a

a = `1/2 ("d"^2y)/("d"x^2)`  .......(3)

From (2) and (3)

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

b = `("d"y)/("d"x) - x ("d"^2y)/("d"x^2)`  .........(4)

Substitute the value of a and b in equation (1)

Equation (1)

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

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

Multiply each term by 2

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

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

⇒ `x^2 ("d"^2y)/("d"x^2) - 2x ("d"y)/(d") + 2y`= 0