Using the substitution u(x) = y + x, solve the differential equation dy/dx = (y + x) ² .

Solution:

Given differential equation:

dy/dx = (y + x) ² and u(x) = y + x

Let

dy/dx = (y + x) ² ……(1)

For convenience let us take

x + y = z……………(2)

Differentiating (2) w.r.t. x, we get

1 + dy/dx = dz/dx

dy/dx = dz/dx -1………(3)

Substituting (3) in (1),

dz/dx -1 = z²

dz/dx = z² + 1

dz / (z² + 1) = dx

Integrating on both the sides,

∫ 1 / (z² + 1) . dz = ∫ 1. dx

tan^-1(z) = x + C

Substituting the value of z, we get

⇒ tan^-1(x + y) = x + C [From (2)]

Hence the required solution is tan^-1(x + y) = x + C.

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Using the substitution u(x) = y + x, solve the differential equation dy/dx = (y + x) ² .

Summary:

For the given differential equation dy/dx = (y + x) ² , substituting u(x) = y + x, we get tan^-1(x + y) = x + C.