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Sum

**Solve the following differential equation.**

`xy dy/dx = x^2 + 2y^2`

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#### Solution

`xy dy/dx = x^2 + 2y^2`

∴ `dy/dx = (x^2 + 2y^2)/(xy) …(i)`

Put y = tx ...(ii)

Differentiating w.r.t. x, we get

`dy/dx = t + x dt/dx` ...(iii)

Substituting (ii) and (iii) in (i), we get

`t +x dt/dx = (x^2 + 2t^2 x^2)/(x(tx))`

∴`t +x dt/dx = (x^2 (1 + 2t^2))/(x^2t)`

∴ `x dt/dx = (1 + 2t^2)/t - t = (1+t^2)/t`

∴ `t / (1+t^2) dt = 1/xdx`

Integrating on both sides, we get

`1/2 int (2t)/(1+t^2) dt = int dx/x`

∴ `1/2 log |1+ t^2| = log|x| + log |c_1|`

∴ log |1 + t^{2} | = 2 log |x| + 2log |c_{1}|

= log |x^{2}| + log |c| …[logc_{1}^{2} = log c]

∴ log |1 + t^{2}| = log |cx ^{2}|

∴ 1 + t^{2} = cx^{2}

∴ `1+ y^2/x^2 = cx^2`

∴ x^{2} + y^{2} = cx^{4}

Concept: Differential Equations

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