Find the general solution for the following differential equation:
$x^{2} d y + y (x + y) d x = 0$

Question

Find the general solution for the following differential equation:x2dy+y(x+y)dx=0

Filo · Accepted Answer

x2dy=−y(x+y)dx⟹dxdy​=−x2y(x+y)​dxdy​=−x2xy​−x2y2​dxdy​=−xy​−(xy​)2The above differential equation is homogeneous.∴ put vx=y........(1)dxdy​=−v−v2.....(2)differentiating equation (1),dxdy​=v+xdxdv​......(3)by equation (2) & (3),v+xdxdv​=−v2−vxdxdv​=−v2−2v⟹v2+2v1​dv=−x1​dxintegrating both sides,∫v2+2v1​dv=−∫x1​dx∫(v2+2v+1)−11​dv=−ln∣x∣+ln∣c∣∫(v+1)2−121​dv=−ln∣x∣+ln∣c∣      using ∫x2−a21​dx=2a1​ln∣x+ax−a​∣+c21​ln∣∣​v+1+1v+1−1​∣∣​=−ln∣x∣+ln∣c∣21​ln∣∣​v+2v​∣∣​=−ln∣x∣+ln∣c∣ln∣∣​v+2​v​​∣∣​=−ln∣x∣+ln∣c∣ln∣∣​v+2​v​​∣∣​+ln∣x∣=ln∣c∣ln∣∣​v+2​xv​​∣∣​=ln∣c∣⟹v+2​xv​​=cfrom equation (1) v=xy​,xy​+2​xxy​​​=csquaring both sides,(xy+2x​)xy​=c2Therefore,general solution:x2y=c2(y+2x)

Question Text	Find the general solution for the following differential equation: $x^{2} d y + y (x + y) d x = 0$
Answer Type	Text solution:1
Upvotes	150

Find the general solution for the following differential equation:
$x^{2} d y + y (x + y) d x = 0$

Text solutionVerified

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