Detailed Explanations for Calculus 2 Equations

Consider the function:
$f : R \to R$
$f (x) = \frac{x}{| x | + 1}$
I already showed that this function is continuous and injective. Maybe this could help? Maybe we can use the intermediate value theorem? The problem is that I don't have an interval [a,b] with a,b $\in$ $R$ . I know that I can consider the limit, but we didn't proved the fact, that I can use the intermediate value theorem for limits.

Addyson Bright 2022-09-17

Multiplied differential equations
How to get x(t),y(t) solutions for "product differential equations" (dotted on t):
$\dot{x} \dot{y} = x y, {\dot{y}}^{2} - {\dot{x}}^{2} = 1;$
we have by solving quadratics
$(2 {\dot{y}}^{2}, 2 {\dot{x}}^{2}) = + 1 \pm \sqrt{1 + 4 x^{2} y^{2}}, - 1 \pm \sqrt{1 + 4 x^{2} y^{2}}$

clasicaacyx 2022-09-17

Two number have a sum of 124 and a difference of 46. Find the humbers

maredilunavy 2022-09-17

Problem:

Use implicit differentiation to compute $\frac{\partial z}{\partial x}$ and $\frac{\partial z}{\partial y}$ of the function $x^{3} + y^{3} + z^{3} - 3 x y z = 0$ .
What I Got:
$3 x^{2} + 0 + 3 z^{2} \cdot \frac{\partial z}{\partial x} - 3 y \cdot \frac{\partial z}{\partial x}$
$\frac{\partial z}{\partial x} (3 z^{2} - 3 y) = - 3 x^{2}$
$\frac{\partial z}{\partial x} = \frac{- 3 x^{2}}{3 z^{2} - 3 y}$
and
$0 + 3 y^{2} + 3 z^{2} \cdot \frac{\partial z}{\partial y} - 3 x \cdot \frac{\partial z}{\partial y}$
$\frac{\partial z}{\partial y} (3 z^{2} - 3 x) = - 3 y^{2}$
$\frac{\partial z}{\partial y} = \frac{- 3 y^{2}}{3 z^{2} - 3 y}$
For what I have got, is this the correct way to do this question?

curukksm 2022-09-16

Solving Coupled Differential Equations
I have the following differential equations, for modeling predator-prey relationships:
$\frac{d x}{d t} = A x - B x y$
$\frac{d y}{d t} = C x y - D y$
Where A, B, C, and D are constants. How could I go about solving this? I've only really worked with basic first order differential equations before, and I've found little to help me figure this out. Any help would be much appreciated.

videosfapaturqz 2022-09-16

How do you find dy/dx for the function $y = \sin (2 x + 4 y)$

Julia Chang 2022-09-16

My first one is express $\frac{d y}{d x}$ in terms of $x$ and $y$ if $x^{2} - 4 x y^{3} + 8 x^{2} y = 20$
what I did is this
$2 x - 4 x (3 y^{2}) (\frac{d y}{d x}) + 4 y^{3} + 8 x^{2} \frac{d y}{d x} + 16 x y = 0$
$- 4 x (3 y^{2} \frac{d y}{d x}) + 8 x^{2} \frac{d y}{d x} = - 2 x - 4 y^{3} - 16$
Finally I got $\frac{d y}{d x} = \frac{2 x - 4 y^{3} - 16}{- 4 (x) (3 y^{2}) + 8 x^{2}}$
My second question is
Find an equation for the tangent line to the graph of $2 x^{2} - 5 y^{2} + x y = 5$ at the point (2,1)
Anyway I simplified the equation to
$\frac{d y}{d x} = \frac{- 4 x - 1}{- 10 y + x}$ plugging in x and y I got $\frac{9}{8}$
so my line is $y - 1 = \frac{9}{8} (x - 2)$ but I am unsure if if I did this correctly.

Gauge Odom 2022-09-16

About differential equations.
1. While solving differential equations, is it okay to find an expression containing y and x (without y′) from which it's rather difficult to express y, for example,
$\frac{\ln^{2} (y)}{y} - \frac{x}{y} = C ?$
2. Does it make sense to try to find x(y) in some problems where it's easier?

geneth1u 2022-09-15

Use substitution to convert the integrals to integrals of rational functions. Then use partial fractions to evaluate the integrals. $\int \frac{d x}{1 + \sqrt{x + 1}}$

Alexus Deleon 2022-09-15

Find partial derivative of $z$ with respect to the partial of $y$ using the result of the chain rule.
$\ln (x^{2} + y^{2}) + x \ln (z) - \cos (x y z) = 3.$
I would use regular implicit differentiation for this problem but what does "using result of the chain rule" mean?

Ivan Buckley 2022-09-15

What is the derivative of $e^{2 x y}$

batejavizb 2022-09-14

Find dydx by implicit differentiation.
x2−4xy+y2=4

My solution:
$\frac{d}{d x} x^{2} - \frac{d}{d x} 4 x y + \frac{d}{d x} y^{2} = \frac{d}{d x} 4$
$2 x - (4 x)^{'} (y) + (4 x) (y)^{'} + 2 y \frac{d y}{d x} = 0$
$2 x - 4 y \frac{d y}{d x} + 4 x \frac{d y}{d x} + 2 y \frac{d y}{d x} = 0$
$\frac{d y}{d x} (- 4 y + 4 x + 2 y) = - 2 x$
$\frac{d y}{d x} = \frac{- 2 x}{4 x - 2 y}$
Correct answer:
$\frac{d y}{d x} = \frac{2 y - x}{y - 2 x}$
I would like to understand why my answer is incorrect. If someone could take a look at my steps and explain to me where I went wrong it would be greatly appreciated!

Milton Anderson 2022-09-14

I need help with Differential equations.
What is the Implicit answer for this differential equation?
$\frac{d y}{d x} = y^{2} - 4$

Kailey Vargas 2022-09-13

Differential equations systems
We have that system of differential equations:
${\begin{cases} x^{'} = - x \\ y^{'} = - 2 y \end{cases}$
I have to solve that system but I only know the method of derive first equation and substitute in the second and get a second order differential equation which I know to solve. But it seems that that method doesn't work here. How can I solve it? I'm beginner at this kind of exercises.

kybudmanqm 2022-09-12

Distinction between "measure differential equations" and "differential equations in distributions"?
Is there a universally recognized term for ODEs considered in the sense of distributions used to describe impulsive/discontinuous processes? I noticed that some authors call such ODEs "measure differential equations" while others use the term "differential equations in distributions". But I don't see a major difference between them. Can anyone please make the point clear?

Darius Nash 2022-09-12

Use a table of integrals to evaluate the following integrals. $\int \frac{d y}{\sqrt{4 - y^{2}}}$

Jazmyn Saunders 2022-09-12

How are Linear Differential Equations formed?
A linear differential equation of second or higher order have more than 1 solutions of $y (y 1, y 2,$ etc.). Hence a common solution is obtained by assuming $y = c 1 y 1 + c 2 y 2 + . . . . .$ Now my questions are:
1. How are linear differential equations formed from general algebraic expressions?
2. Why is it that a linear differential equation of second or higher order contain more than one solution of y that satisfy them?

Pranav Ward 2022-09-11

Fourier and differential equations
Found this problem, just so you know it's my first time using Fourier to solve differential equations.
$f^{″} (x) + f (x) = 3 \cos (2 x)$

IJzerboor07 2022-09-11

Why aren't exact differential equations considered PDE?
Exact differential equations come from finding the total differential from some multivariable function.
In the exact differential equation $M d x + N d y = 0$
M and N are considered to be partial derivatives of some potential function... So why aren't exact differential equations considered PDEs? After all, you're finding the potential function given it's partial derivatives...

Dulce Cantrell 2022-09-11

Systems of Differential Equations and higher order Differential Equations.
I've seen how one can transform a higher order ordinary differential equation into a system of first-order differential equations, but I haven't been able to find the converse. Is it true that one can transform any system into a higher-order differential equation? If so, is there a general method to do so?

When you are dealing with any Calculus 2 homework, it is vital to have a look at the various questions and answers that will help you see whether you are correct in your approach to finding solutions. Even if you are dealing with analytical aspects of Calculus 2, it will be helpful as you are looking at provided equations and learn how the answers relate to original questions and problems specified.

Do not be afraid to take a look at the basic integration and related application if Calculus 2 does not sound clear or start with the Calculus 1 first.

Calculus 2

Calculus 2 Equations: Expert Solutions and More