Question

Solve the given differential equation by using appropriate substitution.

dy/dx = (1-x-y) / (x+y)

Answer 1

Answer on Question #70257 – Math – Differential Equations

Question

Solve the given differential equation by using appropriate substitution:

$$\frac{dy}{dx} = \frac{1 - x - y}{x + y}$$

Solution

Let $g(x) = -x - y(x) \Rightarrow y(x) = -x - g(x) \Rightarrow -1 - \frac{dg(x)}{dx} = \frac{1 + g(x)}{-g(x)} \Rightarrow 1 + \frac{dg(x)}{dx} = \frac{1 + g(x)}{g(x)} \Rightarrow$

$$\Rightarrow 1 + \frac{dg(x)}{dx} = \frac{1}{g(x)} + 1 \Rightarrow \frac{dg(x)}{dx} = \frac{1}{g(x)} \Rightarrow \frac{dg(x)}{dx} g(x) = 1 \Rightarrow \int g'(x) g(x) dx = \int 1 dx$$ $$\int g'(x) g(x) dx = \begin{bmatrix} u = g(x) \\ du = g'(x) dx \end{bmatrix} = \int u du = \frac{u^2}{2} + c_1 = \frac{g^2(x)}{2} + c_1$$ $$\frac{g^2(x)}{2} + c_1 = x + c_2 \text{ or } \frac{g^2(x)}{2} = x + c \Rightarrow \frac{(x + y)^2}{2} = x + c$$

Answer:

$$\frac{(x + y)^2}{2} = x + c$$

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