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Question
Find the general solution of the differential equation: `e^((dy)/(dx)) = x^2`.
Solution
Given differential equation is `e^((dy)/(dx)) = x^2`
Taking log both sides, we get
`(dy)/(dx) loge` = 2 logx
⇒ `(dy)/(dx)` = 2 logx ...[∵ loge = 1]
⇒ dy = 2 logx dx
On integrating both sides, we get
`intdy = 2intlogxdx`
⇒ y = `2int1.logxdx`
⇒ y = `[logx int1dx - int d/(dx) (logx)(int1.dx)dx]`
⇒ y = `2[logx(x) - int1/x (x)dx]` ...[Using integration by parts]
⇒ y = 2[xlogx – x] + C
⇒ y = 2x(logx – 1) + C
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