Differential Equations
Family of Solution Curves — Intersection Points
nta_pyq_2023_apr
Grade 12
Question:
Let $y=y_1(x)$ and $y=y_2(x)$ be solution curves of $\dfrac{dy}{dx}=y+7$ with initial conditions $y_1(0)=0$ and $y_2(0)=1$ respectively. Then the curves $y=y_1(x)$ and $y=y_2(x)$ intersect at
no point
two points
one point
infinite number of points
Step-by-Step Solution
Key Concept: Solve: $y=Ce^x-7$. $y_1(x)=7(e^x-1)$ and $y_2(x)=8e^x-7$. Set equal: $7(e^x-1)=8e^x-7\Rightarrow -e^x=0$, impossible.
$y_1=y_2\Rightarrow e^x=0$: no solution. Zero intersection points.
Correct Answer: 1