5.6 Differential Equations

Tags

Differential equations

Separable

Dimensionally homogeneous

Inhomogeneous

Euler's method

Another application is differential equation, an equation involving a derivative of a function. (HL)

There are two ways to solve a differential equation, numerical method, and analytic method. A general solution of a differential equation is a function with a constant

c

that satisfies the differential equation.

Below regards the analytic method.

Form	Steps
$\frac{dy}{dx}=f(x)$	1. If $f(x)$ is integrable, then simply, $y = f(x) dx$ . 2. Calculate the constant of integration using the particular value.
$\frac{dy}{dx}=f(x)g(y)$ Separable	1. Separate the given equation to $g(y) dy = f(x) dx$ . 2. Integrate both side to obtain $G(y) =F(x) +$ C . Isolate $y$ if possible. 3. Calculate the constant of integration using the particular value.
$\frac{dy}{dx}=f(\frac yx)$ Dimensionally Homogeneous	1. Substitute $y=vx$ . Note that $\frac{dy}{dx}=v+x\frac{dv}{dx}$ 2. Solve the now separable equation in terms of $v$ . Replace $v$ back in terms of $y$ . 3. Calculate the constant of integration using the particular value.
$\frac{dy}{dx}+p(x)y=q(x)$ Inhomogeneous	1. Calculate the integrating factor $I(x)=e^{p(x) dx}$ 2. Multiply both LHS and RHS by the $I$ to obtain the form: $\frac d{dy}[I(x)y]=I(x)q(x)$ 3. Solve the given equation using one of the methods for the above forms. 4. Calculate the constant of integration using the particular value.