Solve the given differential equation subject to the indicated initial condition. (1 + x^4) dy + x(1 + 4y^2) dx = 0, y(1)= 0

Find $\frac{dw}{dt}$ using the appropriate Chain Rule. Evaluate $\frac{dw}{dt}$ at the given value of t. Function: $w=x\sin y,x=e^t,y=\pi -t$ Value: $t=0$

Describe the motion of a particle when the tangential component of acceleration is 0.

Compute the indefinite integral of the following function.
${\displaystyle r\left(t\right)={14}^{t}i+\frac{1}{3+8t}j+\ln 19tk}$
Select the correct choice below and fill in the answer boxes to complete your choice.
A) ${\displaystyle \int r\left(t\right)dt=()i+()j+()k}$
B) ${\displaystyle \int r\left(t\right)dt=()i+()j+()k+C}$

Solutions of the differential equation ${\displaystyle x^2{y}^{″}-4x{y}^{\prime }+6y=0}$.
In one of my test it given to prove that ${\displaystyle x^3\text{and}{x}^2\left|x\right|}$ are linear independent solutions of the differential equation ${\displaystyle x^{2}y{}^{″}-4x{y}^{\prime }+6y=0}$ on R( here x is independent variable).
But according to me it’s Cauchy Euler equation having general solution as ${\displaystyle y=c_1{x}^3+c_2{x}^2}$, where ${\displaystyle c_1}$ and ${\displaystyle c_2}$ are arbitrary constants. How can be ${\displaystyle x^2\left|x\right|}$ a solution of given ODE as I am unable to find its by giving particular values of constants ${\displaystyle c_1\text{and}{c}_2}$? Please help me to solve it.

What is the surface area of the solid created by revolving
$f(x)=\ln x,x\in [2,3]$ around the x axis?

Calculate:
${\int }_0^{\pi }\cos (x)\log ({\sin }^2(x)+1)dx$

How to find the lower and upper bound of a solution of a Cauchy problem?
I am studying some differential equations especially the Cauchy problems, and I encountered this question:
$$(E):\{\begin{array}{r}{y}^{\prime }=ysi{n}^2(y)\\ y(0)=x_0\end{array}$$
They supposed that ${\displaystyle 0<x_0<\pi }$, and I have to find the upper and lower bounds of the solution, knowing that it is a maximal solution. I don't know where to start. I need some kind of help.