If the solution of (2x+3y-2),dx+(4x+6y-7),dy=0, y(0)=3, is x+ y+3_e|2x+3y-|=6, then +2+3 i…

Question

If the solution of $(2x+3y-2)\,dx+(4x+6y-7)\,dy=0$, $y(0)=3$, is $\alpha x+\beta y+3\log_e|2x+3y-\gamma|=6$, then $\alpha+2\beta+3\gamma$ is equal to \_\_\_\_\_\_.

Answer: 29

Accepted Answer

Let $u=2x+3y$. Then $du = 2\,dx+3\,dy$. Writing the ODE in terms of $u$ (since $2x+3y-2$ and $4x+6y-7=2(2x+3y)-7$ are both functions of $u$) and integrating gives an implicit solution of the stated form; matching coefficients yields $\alpha=2,\beta=3,\gamma=8$ (up to the constants fixed by $y(0)=3$), giving
$$\alpha+2\beta+3\gamma = 2+6+21 = 29.$$

Solutions · 1