Solve the following differential equation:

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

Question

Solve the following differential equation:

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

Accepted Answer

> [!TLDR]
> Treat $x$ as a function of $y$: the equation becomes linear, giving $x=Ce^{y}-y-1$.

## Spotting the trick
Written as $\frac{dy}{dx}=\frac{1}{x+y}$ the equation is neither separable nor linear in $y$. But $\frac{dx}{dy}=\frac{1}{dy/dx}$, so flipping the derivative turns it **linear in $x$**.

## Set up
$$(x+y)\frac{dy}{dx}=1 \implies \frac{dx}{dy} = x + y \implies \frac{dx}{dy} - x = y.$$
This is $\frac{dx}{dy}+P(y)x=Q(y)$ with $P=-1$, $Q=y$.

## Integrating factor
$$\mu(y) = e^{\int(-1)\,dy} = e^{-y}.$$

## Multiply and integrate
$$\frac{d}{dy}\big(x\,e^{-y}\big) = y\,e^{-y}.$$
By parts ($u=y,\ dv=e^{-y}dy$),
$$\int y\,e^{-y}\,dy = -y e^{-y} - e^{-y} + C.$$
So $x\,e^{-y} = -y e^{-y} - e^{-y} + C$. …

Q.Solve the following differential equation: $(x+y)\frac{dy}{dx}=1$

Concept understanding — Integrating Factor Method

Integrating Factor Method

The Standard Form

The Integrating Factor

The Solution

A Quick Illustration

Spotting the trick

Set up

Integrating factor

Multiply and integrate

Method: Flip to $\dfrac{dx}{dy}$ when the equation is linear in $x$

Steps

Common Mistakes

Concept and Intuition

Step-by-Step Solution

Concept

Solution

Q.Solve the following differential equation: (x+y)dydx=1(x+y)\frac{dy}{dx}=1(x+y)dxdy​=1

Spotting the trick

Set up

Integrating factor

Multiply and integrate

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Q.Solve the following differential equation: $(x+y)\frac{dy}{dx}=1$

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