Differential Equation

Tags

math

Refers to an equation consisting of \( x, y \), and \( \frac{\,dy}{\,dx} \).

Solving a differential equation (also known as fining the general solution) means getting it into the form \( y = f(x) \). The general approach to this is to get the left hand side consisting only of \( y \) and \( \,dy \) and the right side consisting only of \( x \) and \( \,dx \). This should only be done using multiplication and division operations. Then we integrate both sides individually to cancel out the delta variables.

Finding the General Solution Examples

For example

\begin{align*} \frac{\,dy}{\,dx} &= \sec^2 x \
\,dy &= \sec^2 x \,dx \
\int 1 \,dy &= \int \sec^2 \,dx \
y &= \tan x + c \end{align*}

For example

\begin{align*} \frac{\,dy}{\,dx} &= xy \
\frac{1}{y} \,dy &= x \,dx \
\int \frac{1}{y} \,dy &= \int x \,dx \
\ln y = \frac{1}{2} x ^2 + c \
e^{\ln y} = e^{\frac{1}{2} x^2 + c} \
y = e^{\frac{1}{2} x^2} e^c \end{align*}

For example

\begin{align*} \frac{\,dy}{\,dx} &= y + xy \
\,dy &= y \,dx + xy \,dx \
\frac{\,dy}{y} &= \,dx + x \,dx \
\frac{\,dy}{y} &= (1+x)\,dx \
\int \frac{1}{y} \,dy &= \int (1+x)\,dx \
\ln y &= x + \frac{1}{2} x^2 + c \
e^{\ln y} &= e^{x + \frac{1}{2} x^2 + c} \
y &= e^x + e^{{\frac{1}{2}x}2}+e^c \end{align*}