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dy + dx (x + 2y) (dx - dy) reducible to homogenous differential equation. |
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Answer» dy + dx = (x+2y) (dx - dy) ⇒ dy + (x+2y) dy = (x+2y) dx - dx ⇒ (x+2y+1) dy = (x+2y-1) dx ⇒ dy/dx = \(\frac{x+2y-1}{x+2y+1}\) Let x+2y = z ⇒ 1 + 2 dy/dx = dz/dx ∴ dz/dx = 1 + 2 (z-1/z+1) ⇒ dz/dx = \(\frac{z+1+2z-2}{z+1}\) = 3z-1/z+1 ⇒ z+1/3z-1 dz = dx ⇒ 1/3 ∫ 3z+3/3z-1 dz = dx ⇒ ∫\(\frac{3z-1+4}{3z-1}\) dz = ∫3 dx ⇒ ∫ dz + ∫ 4/3z-1 dz = ∫3 dx ⇒ z + 4/3 log |3z-1| = 3x + c ⇒ (x+2y) + 4/3 log |3(x+2y-1)| = 3x + c. |
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