Calculus 2 Equations: Expert Solutions and More

Use a table of integrals to evaluate the following integrals. $$\int <!--\; \int \; -->2^{y}dy$$

Determinants and Differential Equations Problem
"[Here] we explore a relationship between determinants and solutions to a differential equation. The $3\times <!--\; \times \; -->3$ matrix consisting of solutions to a differential equation and their derivatives is called the Wronskian and, as we will see in later chapters, plays a pivotal role in the theory of differential equations."
This is the question following the description:
"Verify that $y_1(x)=\cos <!--\; \; -->(2x),y_2(x)=\sin <!--\; \; -->(2x),y_3(x)=e^x$ are solutions to the differential equation: $y^{‴}-<!--\; -\; -->y^{″}+4y^{\prime }-<!--\; -\; -->4y=0$, and show that $\{\{y_1,y_2,y_3\},\{y_1^{\prime },y_2^{\prime },y_3^{\prime }\},\{y_1^{″},y_2^{″},y_3^{″}\}\}$ is nonzero on any interval."
Now I'm really just looking for an explanation on how I would show that it is nonzero on any interval. I completed the first part, I believe, by just taking up to the third derivative for each and plugging each set in to verify an identity of 0=0 at the end. I would then put the terms into the matrix which would give me:
$$\{\{\cos <!--\; \; -->(2x),\sin <!--\; \; -->(2x),e^x\},\{-<!--\; -\; -->2\sin <!--\; \; -->(2x),2\cos <!--\; \; -->(2x),e^x\},\{-<!--\; -\; -->4\cos <!--\; \; -->(2x),-<!--\; -\; -->4\sin <!--\; \; -->(2x),e^x\}\}$$
So I've gotten this far and if anyone could point me in the right direction on where to go from here it would be greatly appreciated.

Use a table of integrals to evaluate the following integrals. $$\int <!--\; \int \; -->x\ast <!--\; \ast \; -->2^{x^2}dx$$

Use a table of integrals to evaluate the following integrals. $$\int <!--\; \int \; -->\frac{dy}{y(2y+9)}$$

Evaluate the integrals.
$$2{\int <!--\; \int \; -->}_4^{25}{x}^{\frac{1}{2}}dx$$

Find the sum of the first 85 terms of the arithmetic series.
$$33+40+47+54+...$$

Decouple differential equations
I have a system of two Second Order differential equations
$$r^2\stackrel{¨<!--\; ¨\; -->}{r}-<!--\; -\; -->r^3({\stackrel{\dot{}<!--\; \dot{}\; -->}{\mathrm{\phi <!--\; \phi \; -->}}}^2+{\mathrm{\omega <!--\; \omega \; -->}}^2)=-<!--\; -\; -->GM$$
$$r\stackrel{¨<!--\; ¨\; -->}{\mathrm{\phi <!--\; \phi \; -->}}+2\stackrel{\dot{}<!--\; \dot{}\; -->}{r}(\stackrel{\dot{}<!--\; \dot{}\; -->}{\mathrm{\phi <!--\; \phi \; -->}}+\mathrm{\omega <!--\; \omega \; -->})=0$$
which I am supposed to decouple using the conservation size $(\stackrel{\dot{}<!--\; \dot{}\; -->}{\mathrm{\phi <!--\; \phi \; -->}}+\mathrm{\omega <!--\; \omega \; -->})r^2$ I have shown, that it is indeed a conservation size, as its derivation is r-times the second equation and therefore zero. However I don't know how this is supposed to help me decoupling the two equations.

Power series solution for differential equations
Is the solution to differential equations using power series applicable only to homogeneous differential equations? I mean equations of the form:
$$a_2{\mathrm{\varphi <!--\; \varphi \; -->}}^{″}(x)+a_1{\mathrm{\varphi <!--\; \varphi \; -->}}^{\prime }(x)+\mathrm{\varphi <!--\; \varphi \; -->}(x)=0$$

Solving differential equations "c" value
I have two questions regarding solving differential equations given initial conditions:
1) When do you substitute the initial conditions into the equation to calculate the value of the constant "c". Do you substitute it once you integrate both sides of the differential equations and you get a constant "c"? Or do you substitute the initial conditions after integrating both sides AS WELL AS rearranging the equations to get y in terms of x and c. Using the second method, sometimes you get two values for "c" with only one value being correct.
2) When you solve certain differential equations, you get one side written with "$\pm <!--\; \pm \; -->$" in the front. However, only one equation fits the initial conditions even after you solve for the constant "c". The one that fits is either the one with the "+" or the one with the "−" in the front. How do you justify which one is correct without giving geometric representations of both and then saying "according to graph, this one insert equation is correct". Can you somehow solve without getting the "$\pm <!--\; \pm \; -->$" in the front?

Can these equations be considered as differential equations?
Consider a differential equation with a term containing $y(x_0)$, for example
$$y^{″}-<!--\; -\; -->2y^{\prime }+y=y(x_0)$$
$x_0\in <!--\; \in \; -->\mathbb{R}$ is a constant. My question is, does such equations fall under the category of differential equations? I have never studied any equation with such a term. If its a differential equation, then $y(x_0)$ can be considered a constant coefficient?

associated homogeneous linear differential equations
Can someone please explain how associated homogeneous linear differential equations work with an example?

Euler differential equations
I have following three equations
$$u^{″}-<!--\; -\; -->2u=-<!--\; -\; -->2v$$
$$u(0)=0$$
$$u^{\prime }(1)=0$$
and from these 3 equations I am trying to find u(v).
It looks to me "Cauchy-Euler Differential Equations - Nonhomogeneous case" but I am not sure about that because it is not an exactly Cauchy form. Could you help me to figure out u(v)?

differential equations of second order
How may I solve this differential equations:
$$y^{″}+4y=12x^2-<!--\; -\; -->16x\cos <!--\; \; -->(2x)?$$

Average rate of change?
How would I figure the following problem out?
Find the average rate of change of $g(x)=x^2+3x+7$ from x=5 to x=9My thought is that I would plug in 5 and 9 for the x values to get the y values. And the use the slope formula $\frac{y_2-<!--\; -\; -->y_1}{x_2-<!--\; -\; -->x_1}$

Help with coupled differential equations
I have these two coupled differential equations
$$a\stackrel{¨<!--\; ¨\; -->}{x}+b\stackrel{⃛<!--\; ⃛\; -->}{x}+c\tan <!--\; \; -->(x+y)\stackrel{\dot{}<!--\; \dot{}\; -->}{x}=0,$$
$$a\stackrel{¨<!--\; ¨\; -->}{y}-<!--\; -\; -->b\stackrel{⃛<!--\; ⃛\; -->}{y}-<!--\; -\; -->c\tan <!--\; \; -->(x+y)\stackrel{\dot{}<!--\; \dot{}\; -->}{y}=0$$
with $a,b,c=const.$
Now I basically have no experience at all with more advanced differential equations. Are these even solvable? I guess answering this type of question is already a science to itself, but maybe you can share some intuition under which circumstances one can hope to find a solution? What are possible methods or tricks?

How do you find the derivative of y in the equation ${\displaystyle \ln \left(xy\right)=x+y}$?

Show that if $f$ is continuous on [0,1] with $f(0)=f(1)$, there must exist $x,y\in <!--\; \in \; -->[0,1]$ with $|x-<!--\; -\; -->y|=\frac{1}{2}$ and $f(x)=f(y)$
I've been working on this for a while, and can't seem to figure out where to start. Any hints would be appreciated

I'm stuck on this implicit differentiation question, where I'm requiblack to find the turning points, the equation goes like this $y^3+3x{y}^2-<!--\; -\; -->x^3=3$

The coefficient of $x^6$ in the Maclaurin series expansion of $f(x)=(1+4x^2{)}^{-<!--\; -\; -->\frac{2}{5}}$ is?

Use substitution to convert the integrals to integrals of rational functions. Then use partial fractions to evaluate the integrals. $$\int <!--\; \int \; -->\frac{1}{1+e^x}dx$$