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Solve: x dy/dx + z = x, x dz/dx + y = 0. |
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Answer» \(x\frac{dy}{dx}\) + z = x ----(1) \(x\frac{dz}{dx}\) + y = 0 -----(2) By adding (1) and (2) we get \(x\frac{dy}{dx}\) (y + z) + (y + z) = x ---(3) Let y + z = ω then equation (3) converts to \(x\frac{d\omega}{dx}+\omega=x\) \(\Rightarrow\) \(\frac{d\omega}{dx}+\frac{\omega }{x}=1 \) ∴ I.F = \(e^{\int\frac{1}{x}dx}\) = elogx = x complete solution is, ω x (I.F) = ∫(I.F) x Q dx = ∫\(x\) x 1 dx = \(\frac{x^2}{2}+c\) \(\Rightarrow\) x(y + z) = \(\frac{x^2}{2}+c\) ----- (4) (∵ I.F = x and ω = y + z) Equation (4) represent solution of pair of given differential equations. |
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