1. Linear first-order ODE
2. Exact equation
3. Constant coefficient Second order linear ODE
4. Variable coefficient Second order linear ODE
5. Variation of parameter과 second order linear ODE의 적용
6. Harmonic Oscillation, Resonance
7. First and second order linear ODE
8. Existence and Uniqueness Theorem
9. Laplace transform 1
10. Laplace transform 2
11. Laplace transform 3
12. Laplace transform 4

1. Linear first-order ODE ¶

from 최정환 https://www.youtube.com/watch?v=J2FJVHchcu4&list=PLL3t9Nt4HrfvpCtvSnkH6Uw0rr169Pzkv&index=23

(a)
미분방정식
$\displaystyle y'=f(x)$
의 해는,
$\displaystyle y=\int f(x)dx$
이것은 고등학교 때 배움.

(b)
$\displaystyle y'=y$
$\displaystyle y=ce^x$
easy

(d)
$\displaystyle y'=ay+b$
sol. (i)
$\displaystyle y'=a(y+\frac{b}a)$
$\displaystyle Y=y+\frac{b}a$
$\displaystyle Y'=aY$
sol. (ii)
$\displaystyle y'+(-a)y=b$
$\displaystyle e^{-ax}y'+(-a)e^{-ax}y=be^{-ax}$
좌변은 $\displaystyle =(e^{-ax}y)'$
$\displaystyle e^{-ax}y=-\frac{b}{a}e^{-ax}+c$
$\displaystyle y=-\frac{b}a+ce^{ax}$

(e)
$\displaystyle y'=ay+b(x)$
(d)의 (i)방식으로는 너무 복잡해진다.
(d)의 (ii)방식으로 하면,
$\displaystyle e^{-ax}y'-ae^{-ax}y=b(x)e^{-ax}$
좌변은 $\displaystyle =(e^{-ax}y)'$
$\displaystyle y=e^{ax}\int b(x)e^{-ax}dx$

(f) 가장 일반적인 형태
$\displaystyle y'=a(x)y+b(x)$ or
$\displaystyle y'+a(x)y=b(x)$
$\displaystyle e^{\int a(x)dx}$ 를 곱하면
$\displaystyle e^{\int a(x)dx}y'+a(x)e^{\int a(x)dx}y=b(x)e^{\int a(x)dx}$
좌변이 $\displaystyle \left[e^{\int a(x)dx}y\right]'$
$\displaystyle e^{\int a(x)dx}y=\int\left[b(x)e^{\int a(x)dx}\right]dx$
$\displaystyle y=e^{-\int a(x)dx}\int\left[b(x)e^{\int a(x)dx}\right]dx$

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2. Exact equation ¶

$\displaystyle y'=f(x,y)=\frac{-M(x,y)}{N(x,y)}$
$\displaystyle \Leftrightarrow M(x,y)+N(x,y)y'=0$
$\displaystyle \Leftrightarrow M(x,y)+N(x,y)\frac{dy}{dx}=0$
$\displaystyle \Leftrightarrow M(x,y)dx+N(x,y)dy=0$

is exact if $\displaystyle \exists F(x,y)$ such that $\displaystyle F_x=M$ and $\displaystyle F_y=N$

Thm.
If $\displaystyle M,N,M_y,N_x$ are all continuous, then

$\displaystyle Mdx+Ndy=0$

is exact iff

$\displaystyle M_y=N_x$ .

Pf. $\displaystyle \exists F$ s.t. $\displaystyle F_x=M,F_y=N$
$\displaystyle M_y=F_{xy}=F_{yx}=N_x$
suppose $\displaystyle M_y=N_x$

포기... https://www.youtube.com/watch?v=brZjGvnsNBo&index=24&list=PLL3t9Nt4HrfvpCtvSnkH6Uw0rr169Pzkv

https://www.youtube.com/watch?v=QQlmSfBmTQg&list=PLL3t9Nt4HrfvpCtvSnkH6Uw0rr169Pzkv&index=27

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3. Constant coefficient Second order linear ODE ¶

2nd order linear homogeneous equation with constant coefficients

$\displaystyle Ay''+By'+Cy=0$ (A,B,C 상수)

$\displaystyle y=e^{\lambda x},$ λ 상수
$\displaystyle y'=\lambda e^{\lambda x}$
$\displaystyle y''=\lambda^2 e^{\lambda x}$
그러면 본 식은
$\displaystyle A\lambda^2 e^{\lambda x}+B\lambda e^{\lambda x}+Ce^{\lambda x}=0$
$\displaystyle (A\lambda^2+B\lambda+C)e^{\lambda x}=0$
왼쪽의 $\displaystyle (A\lambda^2+B\lambda+C)$ 을 characteristic equation이라 함.

특성 방정식

characteristic equation differential

case 1. $\displaystyle B^2-4AC>0$
이차방정식이 두 실근 $\displaystyle \lambda_1\ne\lambda_2$ 를 가진다.

생략..

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4. Variable coefficient Second order linear ODE ¶

https://www.youtube.com/watch?v=nWP0AvG1Yqg&list=PLL3t9Nt4HrfvpCtvSnkH6Uw0rr169Pzkv&index=28

$\displaystyle y''+p(x)y'+q(x)y=0$
Suppose a solution $\displaystyle y_1$ is given.
Let $\displaystyle y_2=y_1v(x),\;v(x)\ne C$
Put $\displaystyle y_2=y_1v(x)$ into the given equation.
$\displaystyle (y_1v)''+P(y_1v)'+qy_1v=0$
$\displaystyle (y_1v)'=y_1'v+y_1v'$
$\displaystyle (y_1v){' '}=y_1{' '}v+2y_1'v'+y_1v{' '}$

이하skip

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