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Question
Show that y = ax3 + bx2 + c is a solution of the differential equation \[\frac{d^3 y}{d x^3} = 6a\].
Solution
We have,
\[y = a x^3 + b x^2 + c ...........(1)\]
Differentiating both sides of (1) with respect to x, we get
\[\frac{dy}{dx} = 3a x^2 + 2bx ...........(2)\]
Differentiating both sides of (2) with respect to x, we get
\[\frac{d^2 y}{d x^2} = 6ax + 2b..............(3)\]
Differentiating both sides of (3) with respect to x, we get
\[\frac{d^3 y}{d x^3} = 6a\]
Hence, the given function is the solution to the given differential equation.
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