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