The process of evaluating a definite integral from a set of... | Agrigyan

Notice that the left hand side is a derivative of a product:
$xy'' + y' = \frac{d}{dx}(x y') = 4x$.
Integrating both sides with respect to $x$:
$x y' = \int 4x\,dx = 2x^2 + C_1$.
Dividing by $x$:
$y' = \frac{dy}{dx} = 2x + \frac{C_1}{x}$.
Integrating once more with respect to $x$:
$y = \int \left(2x + \frac{C_1}{x}\right)dx = x^2 + C_1 \ln x + C_2$.

Detailed analysis of options:

• $y = x^3 + C_1 \ln x + C_2$ has $x^3$ instead of $x^2$.
• $y = x^2 + C_1 \ln x + C_2$ is correct.
• $y = e^x + C_1 \ln x + C_2$ is incorrect.
• $y = e^{-x} + C_1 \ln x + C_2$ is incorrect.

The auxiliary characteristic equation is:
$m^2 + 6m + 9 = 0$.
Factoring gives:
$(m + 3)^2 = 0 \implies m_1 = m_2 = -3$ (repeated real roots).
For repeated roots $m = -3$, the general solution is:
$x = (C_1 + C_2 t)e^{-3t}$.

Detailed analysis of options:

• $x = (C_1 + C_2 t)e^{3t}$ has root $+3$ instead of $-3$.
• $x = (C_1 + C_2 t)e^{-3t}$ is correct.
• Root 5 is incorrect.
• Root $-5$ is incorrect.

In numerical analysis, the shift operator $E$ relates to finite difference operators as:
$E = 1 + \Delta$, and $E^{-1} = 1 - \nabla$.
Multiplying the two operators:
$(1 + \Delta)(1 - \nabla) = E \cdot E^{-1} = 1$.
Alternatively: $(1 + \Delta)(1 - \nabla) = 1 - \nabla + \Delta - \Delta\nabla$.
Since $\Delta - \nabla = \Delta\nabla$, the terms cancel, yielding 1.

Detailed analysis of options:

• 0 is incorrect.
• is the correct identity ($E \cdot E^{-1} = 1$).
• 1 is incorrect.
• is incorrect.