Related Formula
L'Hopital's Rule for (0)/(0) form: t → x (f(t))/(g(t)) = t → x (f'(t))/(g'(t))$$\text{L'Hopital's Rule for } \frac{0}{0} \text{ form: } \lim_{t \to x} \frac{f(t)}{g(t)} = \lim_{t \to x} \frac{f'(t)}{g'(t)}$$
Linear DE standard form: (dy)/(dx) + P(x)y = Q(x)$$\text{Linear DE standard form: } \frac{dy}{dx} + P(x)y = Q(x)$$
Core Logic
Evaluate the limit using L'Hopital's rule, differentiating with respect to t$t$ (treating x$x$ as a constant).
t → x ((d)/(dt) (t² y(x) - x² y(t)))/((d)/(dt)(x - t)) = 3$$\lim_{t \to x} \frac{\frac{d}{dt} (t^2 y(x) - x^2 y(t))}{\frac{d}{dt}(x - t)} = 3$$
t → x (2t · y(x) - x² y'(t))/(-1) = 3$$\lim_{t \to x} \frac{2t \cdot y(x) - x^2 y'(t)}{-1} = 3$$
Substitute t = x$t = x$:
(2x · y(x) - x² y'(x))/(-1) = 3$$\frac{2x \cdot y(x) - x^2 y'(x)}{-1} = 3$$
x² y'(x) - 2x y(x) = 3$$x^2 y'(x) - 2x y(x) = 3$$
Rearrange into standard linear differential equation format:
(dy)/(dx) - (2)/(x) y = (3)/(x²)$$\frac{dy}{dx} - \frac{2}{x} y = \frac{3}{x^2}$$
Step 1: Integrating Factor
I.F. = e∫ P dx = e∫ -(2)/(x) dx = e-2ln x = x⁻² = (1)/(x²)$$\text{I.F.} = e^{\int P dx} = e^{\int -\frac{2}{x} dx} = e^{-2\ln x} = x^{-2} = \frac{1}{x^2}$$
Step 2: Solving the Differential Equation
Multiply the entire equation by I.F.:
y · ((1)/(x²)) = ∫ (3)/(x²) · (1)/(x²) dx$$y \cdot \left(\frac{1}{x^2}\right) = \int \frac{3}{x^2} \cdot \frac{1}{x^2} dx$$
(y)/(x²) = ∫ 3x⁻⁴ dx = 3 ( x⁻³-3) + c$$\frac{y}{x^2} = \int 3x^{-4} dx = 3 \left(\frac{x^{-3}}{-3}\right) + c$$
(y)/(x²) = -(1)/(x³) + c$$\frac{y}{x^2} = -\frac{1}{x^3} + c$$
y(x) = cx² - (1)/(x)$$y(x) = cx^2 - \frac{1}{x}$$
Step 3: Using the Boundary Condition
Given y(1) = 2$y(1) = 2$:
2 = c(1)² - (1)/(1) 2 = c - 1 c = 3$$2 = c(1)^2 - \frac{1}{1} \implies 2 = c - 1 \implies c = 3$$
Thus, the function is y(x) = 3x² - (1)/(x)$y(x) = 3x^2 - \frac{1}{x}$.
Step 4: Evaluating the Target Value
Find y(2)$y(2)$:
y(2) = 3(2)² - (1)/(2) = 12 - (1)/(2) = (23)/(2)$$y(2) = 3(2)^2 - \frac{1}{2} = 12 - \frac{1}{2} = \frac{23}{2}$$
The question asks for 2y(2)$2y(2)$:
2y(2) = 2 ((23)/(2)) = 23$$2y(2) = 2 \left(\frac{23}{2}\right) = 23$$
Pattern Recognition
A limit expression resembling Newton's difference quotient applied to a function block almost universally unwraps into a first-order linear differential equation via L'Hopital's rule with respect to the limit dummy variable.
Chapter Mix
Class 12 Maths: Differential Equations
Class 11 Maths: Limits and Derivatives
