# 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 = z^{2}

dz/dx = z^{2} + 1

dz / (z^{2} + 1) = dx

**Integrating on both the sides,**

∫ 1 / (z^{2} + 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.**

## 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.

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