Differential Equations
Homogeneous DE — Substitution
nta_pyq_2024_jan
Grade 12
Question:
If the solution of the differential equation $(2x+3y-2)dx+(4x+6y-7)dy=0$, $y(0)=3$, is $\alpha x+\beta y+3\log_e|2x+3y-\gamma|=6$, then $\alpha+2\beta+3\gamma$ is equal to
Step-by-Step Solution
Key Concept: Note $4x+6y=2(2x+3y)$. Let $t=2x+3y$; then $dt=2\,dx+3\,dy$. Express $dx$ in terms of $dt$ and $dy$ (or $dt/dx$), substitute to get a separable ODE in $t$ and $x$.
Let $t=2x+3y$: $\frac{dt}{dx}=2+3\frac{dy}{dx}=\frac{t-6}{2t-3}$ after substitution. Integrating $\frac{2t-3}{t-6}dt=dx$: $2t+9\ln|t-6|=x+c$. Apply $y(0)=3\Rightarrow t=9$: $c=18+9\ln3-14$. Simplifying: $x+2y+3\ln|2x+3y-8|=6$. So $\alpha=1,\beta=2,\gamma=8$, $\alpha+2\beta+3\gamma=29$.
Correct Answer: 29