Learn Laplace Transform Equations with Plainmath

Solve the following equation?
${\displaystyle (x-2xy-y^2)\frac{dy}{dx}+y^2=0}$

How do you solve the Initial value probelm
${\displaystyle \frac{dp}{dt}=10p(1-p)}$,
${\displaystyle p\left(0\right)=0.1}$
Solve and show that ${\displaystyle p\left(t\right)\to 1\text{as}t\to \mathrm{\infty }}$.

Solving second-order nonlinear ordinary differential equation
${\displaystyle y{}^{″}\left(x\right)=\frac{4}{3}y{\left(x\right)}^3{y}^{\prime }\left(x\right)}$
given that ${\displaystyle y\left(0\right)=1\text{and}{y}^{\prime }\left(0\right)=\frac{1}{3}}$.

Find the solution of the following Differential Equation
${\displaystyle x{y}^{\prime }=2x+2y}$

Suppose we have a mechanical system with 1 degree of freedom, i.e. an ODE
${\displaystyle \left(1\right)\ddot{q}+V^{\prime }\left(q\right)=0}$,
where ${\displaystyle V:R\to R}$ is some smooth function (potential energy). We then easily see that any solution of this equation must satisfy
${\displaystyle \frac{{\dot{q}}^2}{2}+V\left(q\right)=\text{constant}}$
In other words, if we put
${\displaystyle E(q,\dot{q})=\frac{{\dot{q}}^2}{2}+V\left(q\right)}$
(energy), then the image of every solution of (1) must lie in a level set of E.

How can I solve this non-linear differential equation?
${\displaystyle y^{\prime }=1-y^2}$

In Physics, the Simple Harmonic Oscillator is represented by the equation $d^{2}x/d{t}^2=-{\omega }^{2}x$.
By using the characteristic polynomial, you get solutions of the form ${\displaystyle x\left(t\right)=A{e}^{i\omega t}+B{e}^{-i\omega t}}$, I get that you Euler's formula ${\displaystyle e^{i\theta }=\cos \theta +i\sin \theta }$, but I can't seem to find my way all the way to the ''traditional form'' of ${\displaystyle D\cos \omega t+C\sin \omega t}$.

How to solve this differential equation? ${\displaystyle y{}^{″}-\frac{y^{\prime }}{x}=4x^{2}y}$

Solve the Homogenous Differential Equations.
${\displaystyle (x-y\ln y+y\ln x)dx+x(\ln y-\ln x)dy=0}$

Each of the differential equations is of two different types considered in this chapter-separable, linear, homogeneous, Bernoulli, exact, etc. Hence, derive general solutions for each of these equations in two different ways; then reconcile your results.
${\displaystyle \frac{dy}{dx}=3(y+7)x^2}$

Solutions to the equation ${\displaystyle y^{n}y=1}$ for even n

Counterintuitive PDE
After thinking about it for a while and consulting other students, no one seems to be able to find an example of the following:
Given the PDE
${\displaystyle \frac{\partial f}{\partial x}=0\text{on}U=(x,y)\in R^2;y>0,1<x^2+y^2<4}$
I am looking for a solution f that does not only depend on y.
How can this be?!
The exercise is taken form Lees

I have heard about a lot of fancy numerical methods for solving ODEs. I know that there are methods that (assuming sufficient smoothness) asymptotically give a low error, like the Runge-Kutta methods. These estimate the solution in a set of points ${\displaystyle t_0,t_1}$, etc. But what if I want to have a function that is close to the correct solution everywhere, not just in a discrete set of points?
I can extend the numerical solution to a piecewise linear function. This will be a continuous function and it will converge to the correct solution if the step-size goes to zero.
But the error estimate will be poor in most places unless I use a very small step-size, which rather defeats the purpose of using a high-order method. So how does one go about estimating the solution in practice between the ${\displaystyle t_i}$?

Which of the following differential equations is exact?
${\displaystyle (2x+3y)dx+(3x+2y)dy=0}$

${\displaystyle \frac{dy}{dt}+2y=1;y\left(0\right)=\frac{5}{2}}$
when i solve this i get ${\displaystyle \ln (-4)=c}$ now the problem is how to solve ${\displaystyle \ln (-4)}$?

Find the orthogonal trajectory.
${\displaystyle y^2=k{x}^3}$

I was toying with equations of the type ${\displaystyle f(x+\alpha )=f^{\prime }\left(x\right)}$ where f is a real function. For example if ${\displaystyle \alpha =\frac{\pi }{2}}$ then the solutions include the function ${\displaystyle f_{\lambda ,\mu }\left(x\right)=\lambda \cos (x+\mu )}$. Are there more solutions?
On the other hand, if I want to solve the equation for any ${\displaystyle \alpha }$, I can assume a solution of the form ${\displaystyle f\left(x\right)=e^{\lambda x}}$, and find ${\displaystyle \lambda }$ as a complex number that enables me to solve the equation...
I was wondering: is the set of the solutions of dimension 2 because the derivative operator creates one dimension and the operator ${\displaystyle \varphi :f\left(x\right)\to f(x+1)}$ adds another one? Is there some litterature about this kind of equations?

Applications of First Order Differential Equations: A dead body was found within a closed room of a house where the temperature was a constant ${\displaystyle {24}^{\circ }C}$. At the time of discovery, the core temperature of the body was determined to be ${\displaystyle {28}^{\circ }C}$. One hour later, a second measurement showed that the core temperature of the body was ${\displaystyle {26}^{\circ }C}$. The core temperature of the body at the time of death is ${\displaystyle {37}^{\circ }C}$. A. Determine how many hours elapsed before the body was found. B. What is the temperature of the body 5 hours from the death of the victim?

A question asks us to solve the differential equation
${\displaystyle -u\left(x\right)=\delta \left(x\right)}$
with boundary conditions
${\displaystyle u(-2)=0\text{and}u\left(3\right)=0}$ where ${\displaystyle \delta \left(x\right)}$ is the Dirac delta function. But inside the same question, teacher gives the solution in two pieces as ${\displaystyle u=A(x+2)\text{for}x\le 0\text{and}u=B(x-3)\text{for}x\ge 0}$. I understand when we integrate the delta function twice the result is the ramp function R(x). However elsewhere in his lecture the teacher had given the general solution of that DE as
${\displaystyle u\left(x\right)=-R\left(x\right)+C+Dx}$

In the following problems, obtain the differential equations of the given families of curves whose properties are indicated
Parabolas with vertex on the y-axis, with axis parallel to the x-axis, and with distance from the focus to vertex fixed at a.