If for the solution curve y = f(x)$y = f(x)$ of the differential equation (dy)/(dx) + ( x)y = (2 + x)/((1 + 2 x)²)$\frac{dy}{dx} + (\tan x)y = \frac{2 + \sec x}{(1 + 2\sec x)^2}$, x in ((-π)/(2), (π)/(2)), f((π)/(3)) = √(3)10$x \in \left(\frac{-\pi}{2}, \frac{\pi}{2}\right), f\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{10}$, then f((π)/(4))$f\left(\frac{\pi}{4}\right)$ is equal to:

Related Formula

For differential equations where terms are functions of a linear expression like (x+y)$(x+y)$, substitute a new variable t = x+y$t = x+y$ to enable variable separation.

Core Logic

Given the differential equation:

Let x+y = t 1 + (dy)/(dx) = (dt)/(dx) (dy)/(dx) = (dt)/(dx) - 1$x+y = t \implies 1 + \frac{dy}{dx} = \frac{dt}{dx} \implies \frac{dy}{dx} = \frac{dt}{dx} - 1$.

Substituting these terms back:

Step 1: Separating Variables and Integrating

Separating variables:

Multiply numerator and denominator by t$t$:

Let t² = z 2t dt = dz t dt = (dz)/(2)$t^2 = z \implies 2t \, dt = dz \implies t \, dt = \frac{dz}{2}$:

Using partial fractions:

Step 2: Apply the Boundary Condition

Given y(0) = 1$y(0) = 1 \implies$ at x = 0$x = 0$, y = 1 t = 0 + 1 = 1 z = t² = 1$y = 1 \implies t = 0 + 1 = 1 \implies z = t^2 = 1$. Substituting these value constraints into our integral solution:

Thus:

Since z = t² = (x+y)²$z = t^2 = (x+y)^2$, we have:

Step 3: Evaluate at the Target Value

We need to find the value of the function at x = 1√(2)$x = \frac{1}{\sqrt{2}}$:

Pattern Recognition

Sees: Differential equation format y' = f(x+y)$y' = f(x+y)$. Shortcut: A linear argument (x+y)$(x+y)$ strongly implies substituting t=x+y$t=x+y$. The final expression requested matched the functional template (x+y)²$(x+y)^2$ exactly, saving steps from extracting standalone square roots.

Chapter Mix

Class 12 Mathematics: Differential Equations

Related Formula

A first-order linear differential equation in standard form (dx)/(dt) + P(t)x = Q(t)$\frac{dx}{dt} + P(t)x = Q(t)$ is solved using the Integrating Factor:

Core Logic

Let's rearrange the given differential equation into standard linear form:

Step 1: Compute Integrating Factor and General Solution

Here, P(t) = -(2)/(t+1)$P(t) = -\frac{2}{t+1}$ and Q(t) = (t+1)³$Q(t) = (t+1)^3$.

The general solution is given by:

Step 2: Apply Boundary Condition

Given the initial condition x(0) = 2$x(0) = 2$:

Hence, the specific solution curve is:

Step 3: Evaluate at t = 1

Substituting t = 1$t = 1$:

Pattern Recognition

Sees: Linear form hidden under differential grouping coefficients. Shortcut: Always separate terms to identify whether it matches a standard integrating factor structure. Calculating limits row-by-row on factors prevents algebraic grouping mistakes.

Chapter Mix

Class 12 Mathematics: Differential Equations

Related Formula

Core Logic

Let us reorganize the given differential equation:

Dividing both sides by x²$x^2$:

Step 1: Integration Process

Let (y)/(x) = t$\frac{y}{x} = t$. The equation transforms to:

Integrating both sides:

Step 2: Resolving Constant via Boundary Limits

Given boundary state y(1) = (π)/(3)$y(1) = \frac{\pi}{3}$:

Comparing with the given form ((y)/(x)) = ₑ|x| + (α)/(2)$\sin\left(\frac{y}{x}\right) = \log_e|x| + \frac{\alpha}{2}$:

Therefore: α² = 3$\alpha^2 = 3$

Pattern Recognition

Recognize the standard quotient derivative pattern early. Instead of substituting y = vx$y = vx$ mechanically, collapsing the exact differential notation directly drops layout complexities.

Chapter Mix

Class 12 Mathematics: Differential Equations

Related Formula

Using derivative definition limit formats:

Core Logic

Squaring both sides of the structural limit equation to remove root blocks:

Evaluating limits as r arrow x$r \rightarrow x$:

Step 1: Transforming variables

Reorganizing the differential form:

Substitute homogeneous parameters y = vx (dy)/(dx) = v + x(dv)/(dx)$y = vx \implies \frac{dy}{dx} = v + x\frac{dv}{dx}$:

Step 2: Integrating and Boundary Resolution

Integrating both sides:

Given f(1) = 1 x = 1, y = 1 v = 1$f(1) = 1 \implies x = 1, y = 1 \implies v = 1$:

Thus, the solution is:

We need to find a$a$ such that f(a) = 0 y = 0 v = 0$f(a) = 0 \implies y = 0 \implies v = 0$:

This gives a = e(2)/(e)-1 = (2)/(e)$a = e^{\frac{2}{e}-1} = \frac{2}{e}$ via standard tracking bounds. Therefore:

Pattern Recognition

Isolate limit groupings that resemble standard derivative templates (r-x$r-x$ in denominator) to easily transform limits into smooth differential calculus equations.

Chapter Mix

Class 12 Mathematics: Differential Equations

Related Formula

Core Logic

Solving the first differential equation:

Integrating both sides:

Given x(0) = 2$x(0) = 2$, we find c₁ = ln 2$c_1 = \ln 2$. Thus:

Step 1: Solving for y(t)

Solving the second differential equation:

Integrating both sides:

Given y(0) = 1$y(0) = 1$, we find c₂ = 0$c_2 = 0$. Thus: y(t) = e-bt$y(t) = e^{-bt}$

Step 2: Applying the condition

We are given 3y(1) = 2x(1)$3y(1) = 2x(1)$. Substituting our solutions at t=1$t=1$:

Rearranging to group exponential terms:

Step 3: Finding t for x(t) = y(t)

Set the two trajectory solutions equal:

Rearranging gives:

Substitute ea-b = (4)/(3)$e^{a-b} = \frac{4}{3}$ from Step 2:

Taking (4)/(3)$\log_{\frac{4}{3}}$ on both sides yields:

Pattern Recognition

For (dz)/(dt) + kz = 0$\frac{dz}{dt} + kz = 0$, the solution is always an exponential decay z = z₀ e-kt$z = z_0 e^{-kt}$. Directly writing down the parametric forms saves integration steps and moves you instantly to the algebra.

Chapter Mix

Class 12 Maths: Differential Equations