미분방정식5 Reduction to separable variables Special Case $\frac{\mathrm{d}y}{\mathrm{d}x} = f(Ax + By + C)$ $(B \neq 0)$ # If $B=0$, then $\frac{\mathrm{d}y}{\mathrm{d}x} = f(Ax + By + C)$ is separable. Solution $u = Ax + By + C$ $\Rightarrow$ $\frac{\mathrm{d}u}{\mathrm{d}x} = A + B\frac{\mathrm{d}y}{\mathrm{d}x}$ $\frac{\mathrm{d}y}{\mathrm{d}x} = f(Ax + By + C)$ $\Rightarrow$ $\frac{1}{B} \frac{\mathrm{d}u}{\mathrm{d}x} -\frac{A}{B}= f.. 2024. 2. 3. Bernoulli's equation 정의1 $\frac{\mathrm{d}y}{\mathrm{d}x} + P(x)y = f(x)y^{n}$ $(n \in \mathbb{R})$ is called Bernoulli's equation. # If $n = 0$ or $n=1$, then $\frac{\mathrm{d}y}{\mathrm{d}x} + P(x)y = f(x)y^{n}$ is a linear equation. Solution Idea : $u = y^{1-n}$ $\Rightarrow$ $y = u^{\frac{1}{1-n}}$ $\Rightarrow$ $\frac{\mathrm{d}y}{\mathrm{d}x} = \frac{1}{1-n} u^{\frac{n}{1-n}} \frac{\mathrm{d}u}{\mathrm{d}x}$ $.. 2024. 2. 3. Exact equation and Homogeneous function Def 1 A differential expression $M(x, y) \mathrm{d}x + N(x, y) \mathrm{d}y$ is an exact differential in $R = \left\{ (x, y) \in \mathbb{R}^{2} \; | \; a 2024. 2. 3. Separable and Linear equation Def 1 A $1$st order Differential Equation of the form $\frac{\mathrm{d}y}{\mathrm{d} x} = g(x)h(y)$ is said to be separable. Solution In this case, we can easily find the solution by collecting $x$ and $y$ terms, and intergrating those terms. So we get $\frac{1}{h(y)} \mathrm{d}y = g(x) \mathrm{d}x$ $\Rightarrow$ $\int \frac{1}{h(y)} \mathrm{d}y = \int g(x) \mathrm{d}x$ Def 2 A $1$st order DE of.. 2024. 2. 3. Basic definition and local existence of a unique solution Notation WLOG : without loss of generality LHS : Left hand side RHS : Right hand side IVP : Initial value problem BVP : Boundary value problem $a_{n}(x)\frac{\mathrm{d}^{n}y}{\mathrm{d} x^{n}} + a_{n-1}(x)\frac{\mathrm{d}^{n-1}y}{\mathrm{d} x^{n-1}} + \cdots + a_{1}(x)\frac{\mathrm{d}y}{\mathrm{d} x} + a_{0}(x)y = g(x)$ $(*)$ Def 1 Any function $\phi$, defined on an interval $I$ and possessing a.. 2024. 2. 2. 이전 1 다음