Determine the general solution of the following differential equations: 4.1 dy/dx + y/(xlnx) − x = 0 4.2 d²y/dx² + 2dy/dx + 5y = 6x²

The given differential equation is $\frac{dy}{dx} + \frac{y}{x \ln x} - x = 0$.
Rearrange it into the standard form of a first-order linear differential equation, $\frac{dy}{dx} + P(x)y = Q(x)$:

Here, $P(x) = \frac{1}{x \ln x}$ and $Q(x) = x$.

The integrating factor is given by $I(x) = e^{\int P(x) dx}$.
First, calculate $\int P(x) dx$:

Let $u = \ln x$. Then $du = \frac{1}{x} dx$.

Substitute back $u = \ln x$:

Assuming $x > 1$ (so $\ln x > 0$), we have $\ln(\ln x)$.
Now, calculate the integrating factor:

Multiply the standard form of the differential equation by $I(x)$:

The left side is the derivative of the product $y \cdot I(x)$:

Integrate both sides with respect to $x$:

Use integration by parts for $\int x \ln x dx$. Let $u = \ln x$ and $dv = x dx$.
Then $du = \frac{1}{x} dx$ and $v = \frac{x^2}{2}$.

Substitute this back into the equation for $y \ln x$:

Solve for $y$:

The general solution for 4.1 is:

The given differential equation is $\frac{d^2y}{dx^2} + 2\frac{dy}{dx} + 5y = 6x^2$.
This is a second-order linear non-homogeneous differential equation. The general solution is $y = y_c + y_p$, where $y_c$ is the complementary solution and $y_p$ is the particular solution.

Consider the homogeneous equation: $\frac{d^2y}{dx^2} + 2\frac{dy}{dx} + 5y = 0$.
The characteristic equation is $r^2 + 2r + 5 = 0$.
Use the quadratic formula $r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$:

The roots are complex conjugates of the form $\alpha \pm i\beta$, where $\alpha = -1$ and $\beta = 2$.
The complementary solution is:

The non-homogeneous term is $Q(x) = 6x^2$. Since $Q(x)$ is a polynomial of degree 2, we assume a particular solution of the form:

Find the first and second derivatives of $y_p$:

Substitute $y_p$, $