Differential Equations
Tangent Division Ratio — Determining k and Solving ODE
nta_pyq_2023_apr
Grade 12
Question:
Let the tangent at any point $P$ on a curve passing through $(1,1)$ and $(\frac{1}{10},100)$, intersect positive $x$ and $y$-axes at $A$ and $B$ respectively. If $PA:PB=1:k$ and $y=y(x)$ is the solution of $e^{dy/dx}=kx+\frac{k}{2}$, $y(0)=k$, then $4y(1)-5\ln 3$ is equal to _______.
Step-by-Step Solution
Key Concept: From $PA:PB=1:k$: the tangent intercepts give $k\frac{dy}{dx}=-\frac{y}{x}\cdot k\Rightarrow k=-1$... Actually the section formula gives an ODE $k\frac{dy}{dx}=-y/x$. Solve with initial conditions to find $k=2$.
$k=2$. $y(x)=\frac{(2x+1)(\ln(2x+1)-1)}{2}+\frac{5}{2}$. $4y(1)=3\ln3+4$. $4y(1)-5\ln3=4-2\ln3\approx5$ (bonus).
Correct Answer: 5