**Toolbox:**

- First order , first degree DE
- Variable separable : Variables of a DE are rearranged to separate then, ie
- $f_1(x)g_2(y)dx+f_2(x)g_1(y)dy=0$
- Can be written as $ \large\frac{g_1 (y)}{g_2(y)}$$dy=-\large\frac{f_1(x)}{f_2(x)}$$dx$
- The solution is therefore $\int \large\frac{g_1(y)}{g_2(y)}$$dy=-\int \large\frac{f_1(x)}{f_2(x)}$$dx+c$
- A D.E of first order and first degree is said to be homogeneous if it can be put in the form $\large\frac{dy}{dx}=f\bigg(\large\frac{y}{x}\bigg)$$\;or\; \large\frac{dy}{dx}=\frac{f_1(x,y)}{f_2(x,y)}$ Where $f_1$ and $f_2$ are homogeneous functions in x and y.
- To solve we put $y=vx$ and proceed.

Step 1:

$(x^2+y^2)dx+3xydy=0$

$\large\frac{dy}{dx} =\frac{-(x^2+y^2)}{3xy}$ is homogeneous in $x,y$

$y=vx$=> $\large\frac{dy}{dx}$$=v+x \large\frac{dv}{dx}$

Step 2:

The D.E becomes

$v+ x \large\frac{dv}{dx}=\large\frac{-(x^2+v^2x^2)}{3x.vx}$

$v+ x \large\frac{dv}{dx}=\large\frac{-(1+v^2)}{3.v}$

$ x \large\frac{dv}{dx}=-\bigg(\large\frac{1+v^2}{3v}\bigg)$$-v$

$\qquad= \large\frac{-1-v^2-3v^2}{3v}$

$\qquad= \large\frac{-(1+4v^2)}{3v}$

$\large\frac{3v}{1+4v^2}$$dv=\large\frac{-dx}{x}$

Step 3:

The variables are separated

$\int \large\frac{3v}{1+4v^2}$$dv=-\int \large\frac{dx}{x}$$+\log c_1$

$\large\frac{3}{8} \int \large\frac{8v}{1+4v^2}$$dv = - \int\large\frac{dx}{x}$$+\log c_1$

$\large\frac{3}{8}$$ \log (1+4v^2)+\log x=\log c_1$

$(1+4v^2)^{3/8}x=c_1$

Substitute $v=\large\frac{y}{x}$

$\bigg(1+4 \large\frac{y^2}{x^2}\bigg)^{3/8}$$x=c_1$

$(x^2+4y^2)^{3/8}. x^{1/4}=c_1$

$(x^2+4y^2)^3.x^2=c$ is the G.S