# Differential Equations - Class 12 - Mathematics

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## Extra Questions - Differential Equations | NCERT | Mathematics | Class 12

$\left[\left[f(x) g^{\prime \prime}(x)-f^{\prime\prime}(x) g(x)\right] dx\right]$ is equal to

A) $\frac{f(x)}{g^{\prime}(x)}$

B) $f^{\prime}(x) g(x)-f(x) g^{\prime}(x)$

C) $f(x) g^{\prime}(x)-f^{\prime}(x) g(x)$

D) $f(x) g^{\prime}(x)+f^{\prime}(x) g(x)$

The correct answer is **option C**$$
f(x) g^{\prime}(x) - f^{\prime}(x) g(x)
$$
Here's the detailed solution:

If we integrate the given expression: $$ \int\left[f(x) g^{\prime \prime}(x) - f^{\prime \prime}(x) g(x)\right] dx $$ we can separate it into: $$ \int f(x) g^{\prime \prime}(x) dx - \int f^{\prime \prime}(x) g(x) dx $$ Using integration by parts, where $\int u dv = uv - \int v du$, we can solve each part:

For $\int f(x) g^{\prime \prime}(x) dx$:

Let $u = f(x)$ and $dv = g^{\prime \prime}(x) dx$.

Then $du = f^{\prime}(x) dx$ and $v = g^{\prime}(x)$.

Applying integration by parts: $$ f(x) g^{\prime}(x) - \int f^{\prime}(x) g^{\prime}(x) dx $$

For $\int f^{\prime \prime}(x) g(x) dx$:

Let $u = g(x)$ and $dv = f^{\prime \prime}(x) dx$.

Then $du = g^{\prime}(x) dx$ and $v = f^{\prime}(x)$.

Applying integration by parts: $$ g(x) f^{\prime}(x) - \int g^{\prime}(x) f^{\prime}(x) dx $$

Combining and simplifying these:
$$
(f(x) g^{\prime}(x) - \int f^{\prime}(x) g^{\prime}(x) dx) - (g(x) f^{\prime}(x) - \int g^{\prime}(x) f^{\prime}(x) dx)
$$
results in:
$$
f(x) g^{\prime}(x) - f^{\prime}(x) g(x)
$$
Thus, the integrated expression simplifies to and matches with **option C**.

If $y(x)$ is the solution of the differential equation $(x+2) \frac{dy}{dx} = x^{2} + 4x - 9$, where $x \neq 2$ and $y(0) = 0$, then $y(-4)$ is equal to:

A) 2

B) -1

C) 1

D) 0

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If $f(x)$ is invertible and twice differentiable function satisfying $f'(x) = \int_{0}^{f(x)} f^{-1}(t) dt$, $\forall x \in \mathbb{R}$, and $f'(0) = 1$, then $f'(1)$ can be:

(A) $e$ (B) $e^{2}$ (C) $\frac{1}{e}$ (D) $\sqrt{e}$