# Thread: 1st order linear differential equation

1. ## 1st order linear differential equation

find the general solution
$2(2xy+4y - 3)dx + (x+2)^2dy = 0$

find the general solution
$2(2xy+4y - 3)dx + (x+2)^2dy = 0$

$\implies \frac{dy}{dx} + \frac{4}{x+2}y = \frac{6}{(x+2)^2}$

Theorem:
the general solution of $\frac{dy}{dx} + q(x)y = r(x)$ is given by $y = cy_1 + y_2$, where $y_1 = \frac{1}{Q(x)}$ and $y_2 = \frac{1}{Q(x)}\int_0^x r(\tau )Q(\tau ) d\tau$ and $Q(x) = \exp \left[ \int_0^x q(\tau ) d\tau \right]$, and also $y_1(0) = 1$, $y_2(0) = 0$ so that $y(0) = c$, the initial condition.

now, from the given, we have $q(x) = \frac{4}{x+2}$ and $r(x) = \frac{6}{(x+2)^2}$

so $Q(x) = \exp \left[ \int_0^x \frac{4}{\tau +2} d\tau \right] = \exp \left[ 4 \left( {\ln (x+2) - \ln 2} \right) \right] = \frac{(x+2)^4}{16}$

therefore, $y_1 = \frac{16}{(x+2)^4}$

$y_2 = \frac{16}{(x+2)^4} \int_0^x \frac{6}{(\tau +2)^2} \cdot \frac{(\tau + 2)^4}{16} \, d\tau = \frac{6}{(x+2)^4} \int_0^x (\tau +2)^2\, d\tau$

$= \frac{6}{(x+2)^4} \left( {\frac{x^3}{3} + 2x^2 + 2x} \right) = \frac{2}{(x+2)^4} \left( {x^3 + 6x^2 + 6x} \right) = \frac{2x^3 + 12x^2 + 12x}{(x+2)^4}$

therefore, the general solution is $y = c\cdot \frac{16}{(x+2)^4} + \frac{2x^3 + 12x^2 + 12x}{(x+2)^4}$

3. thanks kuya