If y = y(x)$y = y(x)$ satisfies the differential equation 16 ( x + 9√(x)) (4 + 9 + √(x)) y dy = (1 + 2 y) dx$16 \left(\sqrt{x + 9\sqrt{x}}\right) \left(4 + \sqrt{9 + \sqrt{x}}\right) \cos y \, dy = (1 + 2 \sin y) dx$, x > 0$x > 0$ and y(256) = (π)/(2)$y(256) = \frac{\pi}{2}$, y(49) = α$y(49) = \alpha$, then 2 α$2 \sin \alpha$ is equal to:

Related Formula

Core Logic

The differential equation can be grouped using the exact differential logic.

Notice that x⁴ dy + 4x³ y dx = d(x⁴ y)$x^4 dy + 4x^3 y dx = d(x^4 y)$. So, ∫ d(x⁴ y) = ∫ -2 x dx$\int d(x^4 y) = \int -2\sin x dx$.

Step 1: Integrate both sides

Integrating gives:

Since y(x) = f(x)$y(x) = f(x)$, we have x⁴ f(x) = 2 x + C$x^4 f(x) = 2\cos x + C$.

Step 2: Find Integration Constant

Use the condition f((π)/(2)) = 0$f\left(\frac{\pi}{2}\right) = 0$:

So, the solution is: x⁴ y = 2 x$x^4 y = 2\cos x$

Step 3: Final Calculation

Substitute x = (π)/(3)$x = \frac{\pi}{3}$:

Pattern Recognition

Whenever you see polynomial terms mixed with xⁿ dy + n xⁿ⁻¹ y dx$x^n dy + n x^{n-1} y dx$, always condense it into the exact differential d(xⁿ y)$d(x^n y)$ rather than using the standard integrating factor method.

Chapter Mix

Class 12 Maths: Differential Equations

Related Formula

Core Logic

Differentiate both sides with respect to x$x$ using Leibniz rule:

Rearranging yields a perfect square:

Thus, f'(x) = f(x)$f'(x) = f(x)$.

Step 1: Solve the Differential Equation

Integrate (f'(x))/(f(x)) = 1$\frac{f'(x)}{f(x)} = 1$:

To find k$k$, plug x = 0$x = 0$ into the original integral equation:

Since f$f$ is non-negative, f(0) = 5$f(0) = 5$. Thus, k = 5$k = 5$ and f(x) = 5e^x$f(x) = 5e^x$.

Step 2: Calculate Sequence Values

Evaluate f(ln n)$f(\ln n)$ for n=1, 2, …, 625$n=1, 2, \ldots, 625$:

Step 3: Calculate the Mean

Mean of f(ln 1), f(ln 2), , f(ln 625)$f(\ln 1), f(\ln 2), \dots, f(\ln 625)$:

Pattern Recognition

An integral equation with x$x$ as an upper limit immediately demands Leibniz differentiation. If it results in a perfect square differential relation, an exponential solution map natively emerges.

Chapter Mix

Class 12 Maths: Differential Equations

Related Formula

For a linear differential equation (dy)/(dx) + P(x)y = Q(x)$\frac{dy}{dx} + P(x)y = Q(x)$, the integrating factor is e∫ P(x) dx$e^{\int P(x) dx}$.

Core Logic

Rewrite the equation as:

Dividing by (x² - 4)²$(x^2 - 4)^2$ to create an exact differential:

Observe that the LHS is the exact derivative of (y)/(x² - 4)$\frac{y}{x^2 - 4}$:

Integrating both sides:

Step 1: Finding C

The curve passes through (3, 15)$(3, 15)$. Substitute x=3, y=15$x=3, y=15$:

The function is y = (-x² + 12)(x² - 4)$y = (-x^2 + 12)(x^2 - 4)$.

Step 2: Maxima Calculation

To find the local maximum, differentiate and set y' = 0$y' = 0$:

Critical points at x = 0$x = 0$ and x = ± 2√(2)$x = \pm 2\sqrt{2}$. Given domain x > 2$x > 2$, the valid critical point is x = 2√(2)$x = 2\sqrt{2}$. Check double derivative or nature: It produces a maximum at x = 2√(2)$x = 2\sqrt{2}$.

Substitute x² = 8$x^2 = 8$ into the function for the maximum value:

Pattern Recognition

Any differential equation framed like v u' - u v' = K$v u' - u v' = K$ hints directly at quotient rule exact differentials. Identifying this skips the entire linear integrating factor calculation.

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Class 12 Maths: Differential Equations Class 12 Maths: Application of Derivatives

Related Formula

Newton-Leibniz Rule for differentiating under the integral sign:

Core Logic

Limits: when t = 0, y = 0$t = 0, y = 0$; when t = 36, y = x$t = 36, y = x$.

Step 1: Differentiating with respect to x

Step 2: Integration for Function

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.

So, f(x) = c x²$f(x) = c x^2$.

Step 3: Point Substitution

Passes through (2, 1)$(2, 1)$:

Thus, y = (x²)/(4)$y = \frac{x^2}{4}$. Passes through (-4, β)$(-4, \beta)$:

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