Related Formula
Newton-Leibniz Rule for differentiating under the integral sign:
(d)/(dx) ( ∫u(x)v(x) f(t) dt ) = f(v(x))· v'(x) - f(u(x))· u'(x)$$\frac{d}{dx} \left( \int_{u(x)}^{v(x)} f(t) dt \right) = f(v(x))\cdot v'(x) - f(u(x))\cdot u'(x)$$
Core Logic
Put (tx)/(36) = y ⇒ (x)/(36) dt = dy ⇒ dt = (36)/(x) dy$$\text{Put } \frac{tx}{36} = y \Rightarrow \frac{x}{36} dt = dy \Rightarrow dt = \frac{36}{x} dy$$
Limits: when t = 0, y = 0$t = 0, y = 0$; when t = 36, y = x$t = 36, y = x$.
∫₀^x f(y) (36)/(x) dy = 4α f(x)$$\int_0^x f(y) \frac{36}{x} dy = 4\alpha f(x)$$
∫₀^x f(y) dy = (α x)/(9) f(x)$$\int_0^x f(y) dy = \frac{\alpha x}{9} f(x)$$
Step 1: Differentiating with respect to x
f(x) = (α)/(9) ( f(x) + x f'(x) )$$f(x) = \frac{\alpha}{9} \left( f(x) + x f'(x) \right)$$
f(x) (1 - (α)/(9)) = (α x)/(9) f'(x)$$f(x) \left(1 - \frac{\alpha}{9}\right) = \frac{\alpha x}{9} f'(x)$$
(f'(x))/(f(x)) = ( (9)/(α) - 1 ) (1)/(x)$$\frac{f'(x)}{f(x)} = \left( \frac{9}{\alpha} - 1 \right) \frac{1}{x}$$
Step 2: Integration for Function
ln f(x) = ( (9)/(α) - 1 ) ln x + ln c$$\ln f(x) = \left( \frac{9}{\alpha} - 1 \right) \ln x + \ln c$$
f(x) = c x((9)/(α) - 1)$$f(x) = c x^{\left(\frac{9}{\alpha} - 1\right)}$$
Given y = f(x)$y = f(x)$ is a standard parabola (typically y² = 4ax$y^2 = 4ax$ or x² = 4ay$x^2 = 4ay$). From the polynomial form, it implies degree 2.
(9)/(α) - 1 = 2 ⇒ (9)/(α) = 3 ⇒ α = 3$$\frac{9}{\alpha} - 1 = 2 \Rightarrow \frac{9}{\alpha} = 3 \Rightarrow \alpha = 3$$
So, f(x) = c x²$f(x) = c x^2$.
Step 3: Point Substitution
Passes through (2, 1)$(2, 1)$:
1 = c(2)² ⇒ 4c = 1 ⇒ c = (1)/(4)$$1 = c(2)^2 \Rightarrow 4c = 1 \Rightarrow c = \frac{1}{4}$$
Thus, y = (x²)/(4)$y = \frac{x^2}{4}$.
Passes through (-4, β)$(-4, \beta)$:
β = ((-4)²)/(4) = (16)/(4) = 4$$\beta = \frac{(-4)^2}{4} = \frac{16}{4} = 4$$
β^α = 4³ = 64$$\beta^\alpha = 4^3 = 64$$
Pattern Recognition
Integral equations of the form ∫ f(tx)dt$\int f(tx)dt$ inherently resolve by standardizing the boundary variable (tx → y$tx \to y$) and differentiating, revealing an underlying homogeneous differential equation.
Chapter Mix
Class 12 Maths: Differential Equations
Class 11 Maths: Conic Sections
