Answer to Question #246939 in Differential Equations for Jac

Question

Solve the differential equation by substitution suggested by equation. Show complete solution.

(x + 2y - 1)dx + 3(x + 2y)dy = 0

Answer 1

Solution:

$(x + 2y - 1)dx + 3(x + 2y)dy = 0$ ...(i)

Put $x+2y=v$

$\Rightarrow dx+2dy=dv$

Using these, (i) becomes,

$(v-1)(dv-2dy)+3vdy=0 \\\Rightarrow vdv-2vdy-dv+2dy+3vdy=0 \\\Rightarrow (v-1)dv+(v+2)dy=0 \\\Rightarrow (1-v)dv=(v+2)dy \\\Rightarrow \dfrac{1-v}{v+2}dv=dy$

$\\\Rightarrow \dfrac{1}{v+2}dv-\dfrac{v}{v+2}dv=dy \\\Rightarrow \dfrac{1}{v+2}dv-\dfrac{v+2-2}{v+2}dv=dy \\\Rightarrow \dfrac{1}{v+2}dv-dv+\dfrac{2}{v+2}dv=dy \\\Rightarrow \dfrac{3}{v+2}dv-dv=dy$

On integrating both sides,

$3 \ln|v+2|-v+c=y \\\Rightarrow 3 \ln|x+2y+2|+x+2y+c=y \\\Rightarrow 3 \ln|x+2y+2|+x+y+c=0$

which is required solution.