Explore Laplace Transform Equations

Implicit differentiation question
Given: $\frac{y}{x - y} = x^{2} + 1$

hionormf 2022-01-22

How does a harmonic oscillator with nonlinear damping behave?
$\ddot{x} + c \dot{x} + x = 0$
with positive c, the amplitude of the oscillations decays exponentially when $c < 2$ . If it is higher than 2, the system fails to oscillate at all and is said to be overdamped.

zgribestika 2022-01-22

$\dot{x} (t) = (1 - \frac{x (t)}{E_{0}}) x (t) - a y (t)$

William Cleghorn 2022-01-22

General solution to a differential equation
Is a differential equation still having a general solution even if the differential equation have a singular solution?
$\frac{d y}{d x} = x y^{\frac{1}{2}}$
The solution:
$y = {(\frac{1}{4} x^{2} + c)}^{2}$
But also this singular solution (there is not a constant to obtain it from the above, but is a solution)
$y = 0$
Is this function correct named as a general solution?:
$y = {(\frac{1}{4} x^{2} + c)}^{2}$

Mary Keefe 2022-01-22

What kind of ODE is this and how to solve it?
$(2 x - 4 y + 6) d x + (x + 4 y - 3) d y = 0$
I converted it to this form
$\frac{d y}{d x} = \frac{2 x - 4 y + 6}{x + 4 y - 3}$ in the hope I can use $z = \frac{y}{x}$ substitution for homogeneous ODEs but because of constants 6 and 3 I can't.

Chris Cruz 2022-01-22

Solving an ordinary differential equation:
Prove if $y^{'} (t) + 3 y (t) = 6 t + 5, y (0) = 3$ , then $y (t) = 2 e^{- 3 t} + 2 t + 1$ .

Joan Thompson 2022-01-22

If $f (x) = f^{'} (x) + f^{″} (x)$ then show that $f (x) = 0$
A real-valued function f which is infinitely differentiable on [a.b] has the following properties:
- $f (a) = f (b) = 0$
- $f (x) = f^{'} (x) + f^{″} (x) \forall x \in [a, b]$
Show that $f (x) = 0 \forall x \in [a . b]$

stop2dance3l 2022-01-22

Find the general solution of the differential equation,
$(x + \frac{1}{x}) \frac{d y}{d x} + 2 y = 2 {(x^{2} + 1)}^{2}$
So far, I've divided both sides by $x + \frac{1}{x}$ and integrated $\frac{2 y}{x + \frac{1}{x}}$ to get $y \ln (x^{2} + 1)$ but have no idea where to go from here.

regatamin2 2022-01-22

A-stability of Heun method for ODEs
I'm trying to determine the stability region of the Heun method for ODEs by using the equation $y' = k y$ , where k is a complex number.

maduregimc 2022-01-22

Why the ODE $x^{'} (t) = x^{2} (t)$ doesn't have solutions if $x (0) = x_{0} \neq 0$

Betsy Rhone 2022-01-21

Integrating a product, one factor a derivative
$\int \frac{d x (t)}{d t} x {(t)}^{2} d t$

Dowqueuestbew1j 2022-01-21

A linear differential equation
$x^{″} + 2 c x^{'} + (\frac{2}{{\cos h}^{2} t} - 1) x = 0$
where c is a constant.

interdicoxd 2022-01-21

Using an integrating factor to solve an ODE
$\frac{d c}{d t} = 1 - c - a c$

untchick04tm 2022-01-21

Solving Differential Equation
$(t + 2) d x = 2 x^{2} d t$

Mary Keefe 2022-01-21

A simple question about the solution of homogeneous equation to this differential equation
Given that $t, 1 + t, t^{2}, - t$ are the solutions to $y^{‴} + a (t) y^{″} + b (t) y^{″} + c (t) y = d (t)$ , what is the solution of homogeneous equation to this differential equation? What i have done is tried the properties of linear differential equation that
$L (t) = L (1 + t) = L (t^{2}) = L (- t) = d (t)$ so the homogeneous solution should be independent and i claim that $1, t, t^{2}$ should be the solution. However, i am not sure hot can i actually conclude that these are the solutions? It seems that it can be quite a number of sets of solution by the linearity.

burkinaval1b 2022-01-21

Second derivative using implicit differentiation with respect to x of $x = \sin y + \cos y$
$1 = \cos y \frac{d y}{d x} - \sin y \frac{d y}{d x}$
$1 = \frac{d y}{d x} (\cos y - \sin y)$
$\frac{d y}{d x} = \frac{1}{\cos y - \sin y}$

Mary Reyes 2022-01-21

Let $P_{n}$ be group of all permutations of the set ${1, \dots, n}$ . I have trouble with construction a Lipschitz function $f : P_{n} \to R$ .

Zerrilloh6 2022-01-21

Third order Cauchy-Euler differential equation
Solve: $x^{3} y - x^{2} y + 2 x y - 2 y = x^{3}$

Agohofidov6 2022-01-21

Particular solution to $y^{″} - 3 y^{'} + 2 y = 2 e^{x}$

compagnia04 2022-01-21

Finding all functions f satisfying
$f^{'} (t) = f (t) + \int_{a}^{b} f (t) d t$

If you came across the necessity of Laplace transform, it is most likely that you are coming from a mechanical engineering or electrical background. The concept is used to solve differential equations, which is why it is vital to consider Laplace transform examples as you are looking through the questions and connect the dots as the equations are being approached. Remember to look through our list of answers as these will help you to address various Laplace transform problems and find solutions to complex Laplace transform questions as you are dealing with your Laplace transform equation homework.

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