Solcve: ddxdydx=y(logy–logx+1)

Question

Solcve: `x ("d"y)/("d"x) = y(log y – log x + 1)`

Sum

Solution

Given that: `x ("d"y)/("d"x) = y(log y – log x + 1)`

⇒ `x ("d"y)/("d"x) = y[log(y/x) + 1]`

⇒ `("d"y)/("d"x) = y/x[log(y/x) + 1]`

Since, it is a homogeneous differential equation.

∴ Put y = vx

⇒ `("d"y)/("d"x) = "v" + x * "dv"/"dx"`

∴ `"v" + x * "dv"/"dx" = "vx"/x[log("vx"/x) + 1]`

⇒ `"v" + x * "dv"/"dx" = "v"[log "v" + 1]`

⇒ `x * "dv"/"dx" = "v"[log "v" + 1] - "v"`

⇒ `x * "dv"/"dx"` = v ....[log v + 1 – 1]

⇒ `x * "dv"/"dx" = "v" * log "v"`

⇒ `"dv"/("v"log"v") = "dx"/x`

Integrating both sides, we get

`int "dv"/("v"log"v") = int "dx"/x`

Put log v = t on L.H.S.

`1/"v" "dv"` = dt

∴ `int "dt"/"t" = int "dx"/x`

`log|"t"| = log|x| + log"c"`

⇒ `log|log "v"| = log x"c"`

⇒ log v = xc

⇒ `log(y/x)` = xc

Hence, the required solution is `log(y/x)` = xc.

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Methods of Solving Differential Equations> Homogeneous Differential Equations

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Q 33 Q 32 Q 34

Chapter 9: Differential Equations - Exercise [Page 195]

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NCERT Exemplar Mathematics Exemplar [English] Class 12

Chapter 9 Differential Equations
Exercise | Q 33 | Page 195

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