Equations

Bernoulli Differential Equations If the differential equation is in the form of $$ \begin{align} y^{\prime}+p(x)y=q(x)y^n \end{align} $$ $$ \begin{align} y^{\prime}+p(x)y=q(x)y^n \end{align} $$ If $n=0$, then the differential equation is linear differential equation: $$ \begin{aligned} y^{\prime}+p(x)y=q(x) \end{aligned} $$ $$ \begin{aligned} y^{\prime}+p(x)y=q(x) \end{aligned} $$ If $n=1$, then the differential equation is separable differential equation: $$ \begin{aligned} y^{\prime}+p(x)y &=q(x)y\\ \dfrac{dy}{y} &= (q(x)-p(x))dx \end{aligned} $$ $$ \begin{aligned} y^{\prime}+p(x)y &=q(x)y\\ \dfrac{dy}{y} &= (q(x)-p(x))dx \end{aligned} $$ If $n \neq 1, 0$, then it is said to be Bernoulli’s Differential Equation in $y$. The differential equation is neither separable, homogenous nor linear differential equation for $n \neq 0, 1$. ...

Consider the initial value problem (IVP) $$ \begin{aligned} x=x(t), x_0 = x(t_0) \text{ for } t \geq t_0 \end{aligned} $$ $$ \begin{aligned} x=x(t), x_0 = x(t_0) \text{ for } t \geq t_0 \end{aligned} $$ where $$ \begin{aligned} (x, t) : \vert t - t_0\vert < a, \vert x - x_0\vert < b \end{aligned} $$ $$ \begin{aligned} (x, t) : \vert t - t_0\vert < a, \vert x - x_0\vert < b \end{aligned} $$ this means that the pairs $(x, t)$ satisfies the inequalities $\vert t - t_0\vert < a, \vert x - x_0\vert < b$ for $a,b > 0$. For the sake of understanding we can say $t_0$ is initial time and $x_0$ is initial position. Therefore $(x_0, t_0)$ or $(x(t_0), t_0)$ is initial condition. In this case $t_0$ is independent variable and $x$ is dependent variable (depends on $t$). ...

Definitions As we discussed earlier in here differential equations widely used when we need to model real-world actions. For instance, velocity, acceleration, heat on a surface etc. Remember that, derivatives measure how a quantity changes, so a differential equation describes a relationship between a quantity and its rate of change. Let us leave these intuitive definitions aside and focus on rigorous definitions of differential equations. $\textbf{\small Definition 1.1.:} $ (Differential Equations) An equation containing the derivatives of one or more dependent variables with respect to one or more independent variables is said to be a differential equation, shortly DE. ...

Why to use differential equations I do not and will not like giving plain formula or technic to solve problem, rather I want to learn the truth as Faust did. Faust was extremely successful person, he was some kind of researcher, he read and read. However, he was not satisfied with the position of his life. It seemed meaningles and pointless to him. One day he made a pact with Mephistoteles. The deal was pretty straightforward: in exchange for his soul he will gain unlimited knowledge and hedonic pleasures. If Faust had been a real person, he would have hated plain formulas given without reasoning. Therefore, we will not use any formula, theorem or lemma without proving it beforehand. I have briefly discussed Faust here. ...