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Question
y2 = (x + c)3 is the general solution of the differential equation ______.
Solution
y2 = (x + c)3 is the general solution of the differential equation `bb((dy)/(dx) = 3/2 (root(3)(y)))`.
Explanation:
⇒ y2 = (x + c)3
Differentiating w.r.t. 'x',
`2y * (dy)/(dx) = 3.(x + c)^2`
⇒ `(dy)/(dx) * (2y)/3 = (x + c)^2`
Taking a cube on both sides, we get
`[(x + c)^2]^3 = ((2y)/3 * (dy)/(dx))^3`
⇒ `(y^2)^2 = (8y^2)/27 xx ((dy)/(dx))^3`
⇒ `y^4 xx 27/(8y^3) = ((dy)/(dx))^3`
⇒ `((dy)/(dx))^3 = (27 y)/8`
Taking the cube root of both sides we get,
`(dy)/(dx) = 3/2 root(3)(y)`
This is the required differential equation.
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