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प्रश्न
Solve:
(x + y) dy = a2 dx
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उत्तर
(x + y) dy = a2 dx
∴ `dy/dx = a^2/(x+y)` ...(i)
Put x + y = t ...(ii)
∴ y = t - x
Differentiating w.r.t. x, we get
∴ `dy/dx = dt /dx -1` ....(iii)
Substituting (ii) and (iii) in (i), we get
`dt/dx -1 = a^2/t`
∴ `dt/dx = a^2/t + 1`
∴ `dt/dx = (a^2+t)/t`
∴ `t/(a^2+t) dt = dx`
Integrating on both sides, we get
`int ((a^2+t) - a^2)/(a^2+ t) dt = int dx`
∴ `int 1 dt- a^2int 1/(a^2+t) dt = int dx`
∴ t - a2 log |a2 + t| = x + c1
∴ x + y - a2 log |a2 + x + y| = x + c1
∴ y - a2 log |a2 + x + y| = c1
∴ y - c1 = a2 log |a2 + x + y|
∴ `y/a^2 - c_1/a^2 = log |a^2 + x + y|`
∴ `a^2 + x + y = e^(a^(y/2). e^(a^((-c1)/2)`
∴ `a^2 + x + y = ce^(a^(y/2) ` … `[ c =e^(a^((-c1)/2)]]`
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