Define the term homogeneous first order ordinary differential equation. Hence solve x^2 y' + xy = x2 + y

Are the following are linear equation?
${\displaystyle 1.\sin \left(x\right)\frac{dy}{dx}-3y=0}$
${\displaystyle 2.\frac{d^{2}y}{{dx}^2}+\sin \left(x\right)\frac{dy}{dx}=\cos \left(x\right)}$

Suppose that a dose of ${\displaystyle x_0}$ (initial) gram of a drug is injected into the bloodstream. Assume that the drug leaves the blood and enters the urine at a rate proportional to the amount of drug present in the blood. In addition, assume that half of the drug dose has entered the urine after 0.75 hour. Find the time at which the amount of drug in the blood stream is 5% of the original drug dose ${\displaystyle x_0}$ (initial).
I have the first order differential equation. It's finding ${\displaystyle t}$ that is the problem.

Write an equivalent first-order differential equation and initial condition for y.
${\displaystyle y=-4+{\int }_1^x(4t-y\left(t\right))dt}$
What is the equivalent first-order differential equation?
What is the initial condition?

Solve the equation separable, linear, bernoulli, or homogenous
${\displaystyle 1.\frac{dy}{dx}=\frac{x^4+4x{y}^2}{2x^3+x^{2}y+y^3}}$
${\displaystyle 2.\left(e^{-y}\cos \left(x\right)\right)y^{\prime }=x^4+6x^2{y}^3}$
${\displaystyle 3.y^{\prime }=\frac{y+y{x}^3}{x+x^2}\cos \left(\frac{x^2}{y^2}\right)}$

If I have a differential equation $y^{\prime }(t)=Ay(t)$ where A is a constant square matrix that is not diagonalizable(although it is surely possible to calculate the eigenvalues) and no initial condition is given. And now I am interested in the fundamental matrix. Is there a general method to determine this matrix? I do not want to use the exponential function and the Jordan normal form, as this is quite exhausting. Maybe there is also an ansatz possible as it is for the special case, where this differential equation is equivalent to an n-th order ode. I saw a method where they calculated the eigenvalues of the matrix and depending on the multiplicity n of this eigenvalue they used an exponential term(with the eigenvalue) and in each component an n-th order polynomial as a possible ansatz. Though they only did this, when they were interested in a initial value problem, so with an initial condition and not for a general solution.
I was asked to deliver an example: so $y^{\prime }(t)=\left(\begin{array}{cc}3& -4\\ 1& -1\end{array}\right)y(t)$ If somebody can construct a fundamental matrix for this system, than this should be sufficient

I've tried many times to reach the solution of a first order differential equation (of the last equation) but unfortunately I couldn't. Could you please help me to know how did he get this solution.
The equation is:
${\displaystyle \frac{dX}{dt}=k_f-(k_f+k_b)X}$
The author assumed that
${\displaystyle X=0}$ at ${\displaystyle t=0}$
Then, he said in his first publication the solution is given as:
${\displaystyle X=\frac{k_f}{(k_f+k_b)}\{1-\exp [-(k_f+k_b)t]\}}$
I agree with him about the above equation as I am able to reach this form by using the normal solution of the first order differential equation.
In another publication, the same author dealt with the same equations, but when he wrote the solution of the first order differential equation, he wrote it in another form which I could't reach it and I don't know how did he reach it. This form is given as:
${\displaystyle X=\frac{k_f}{(k_f+k_b)}\{1-\exp [-(k_f+k_b)t]\}+X_0\exp [-(k_f+k_b)t]}$ and he said ${\displaystyle X_0}$ was taken to be zero in the first publication.
Can any of you help how did the author get this solution please?

For every differentiable function $f:\mathbb{R}\to \mathbb{R}$, there is a function $g:\mathbb{R}\times \mathbb{R}\to \mathbb{R}$ such that $g(f(x),f^{\prime }(x))=0$ for every $x$ and for every differentiable function $h:\mathbb{R}\to \mathbb{R}$ holds that
being true that for every $x\in \mathbb{R}$, $g(h(x),h^{\prime }(x))=0$ and $h(0)=f(0)$ implies that $h(x)=f(x)$ for every $x\in \mathbb{R}$.
i.e every differentiable function $f$ is a solution to some first order differential equation that has translation symmetry.