Solutions

General Form

General form for an nth-order ODE with $x$ independent and $y$ dependent can be expressed as

$$F(x,y,\frac{dy}{dy},\frac{d^{2}y}{d{x}^2},\cdots ,\frac{d^{n}y}{d{x}^n})=0$$

Can be rewritten as

$$\frac{d^{n}y}{d{x}^n}=f(x,y,\frac{dy}{dy},\frac{d^{2}y}{d{x}^2},\cdots ,\frac{d^{n-1}y}{d{x}^{n-1}})$$

Explicit Solution

• $\varphi (x)$ is a solution if it satisfies the equation when substituted in for $y(x)$
• Can express the dependent variable as a single function of the independent variable

Implicit Solution

• $G(x,y)=0$ is an implicit solution on interval $I$ if it defines one or more explicit solutions on $I$
• Verify by implicitly differentiating $G(x,y)$ and comparing to $\frac{dy}{dx}$

General Solution

• Solutions to a first order DE $F(x,y,y^{\prime })=0$ are normally a family of solutions $G(x,y,c)=0$ where $c$ is an arbitrary constant
• Solutions to a n-th order DE $F(x,y,y^{\prime },\cdots ,y^{(n)})=0$ are normally a family of solutions $G(x,y,c_1,\cdots ,c_n)=0$ containing $n$ arbitrary constants $c_1,c_2,\cdots ,c_n$