Differential Equations
Differential Equations
nta_pyq_2025_apr
Grade 12
Question:
Let a curve $y = f(x)$ pass through the points $(0,5)$ and $(\log_e 2, k)$. If the curve satisfies the differential equation $2(3+y)e^{2x}\,dx-(7+e^{2x})\,dy = 0$, then $k$ is equal to
Step-by-Step Solution
Key Concept: Separate variables: $\dfrac{dy}{3+y} = \dfrac{2e^{2x}}{7+e^{2x}}dx$; integrate both sides to get $\ln|3+y| = \ln|7+e^{2x}|+C$, then use the initial condition $(0,5)$.
Separating: $\dfrac{dy}{3+y} = \dfrac{2e^{2x}}{7+e^{2x}}dx$.
$\ln|3+y| = \ln|7+e^{2x}|+C_0 \Rightarrow 3+y = A(7+e^{2x})$.
At $(0,5)$: $8 = 8A \Rightarrow A=1$. So $y = e^{2x}+4$.
$$k = y(\ln 2) = e^{2\ln 2}+4 = 4+4 = 8.$$
Correct Answer: 3