Question

(2x+y+3)×(dy/dx)=(x+2y+1)

Answer 1

Answer on Question # 79720 – Math – Differential Equations

Question

Solve the differential equation

$$(2x + y + 3) \times (\frac{dy}{dx}) = (x + 2y + 1)$$

Solution

$$(2x + y + 3) \times (\frac{dy}{dx}) = (x + 2y + 1)$$ $$\frac{dy}{dx} = \frac{x + 2y + 1}{2x + y + 3}$$ $$\text{Or, } (2x + y + 3) \, dy = (x + 2y + 1) \, dx$$

Let $M = (x + 2y + 1)$ and $N = (2x + y + 3)$.

Now, $\frac{\partial M}{\partial x} = 1$ and $\frac{\partial N}{\partial y} = 1$.

So, it is an exact differential equation.

Now, $\int M \, dx = \int (x + 2y + 1) \, dx = \frac{x^2}{2} + 2xy + x + p$, where $p = p(y)$.

Similarly, $\int N \, dy = \int (2x + y + 3) \, dy = 2xy + \frac{y^2}{2} + 3y + q$, where $q = q(x)$.

Here $p$ and $q$ are integration constants.

Combining two expressions the solution is

$$\frac{x^2}{2} + 2xy + x + \frac{y^2}{2} + 3y = c,$$

where $c$ is constant.

Answer: $\frac{x^2}{2} + 2xy + x + \frac{y^2}{2} + 3y = c$, where $c$ is constant.

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