공수1 03년도 중간 기출 및 해답

Problem

 

1. $\frac{dy}{dx}=\frac{e^{2x-y^2}}{2y},y(0)=0$ 을 풀어라.

2. Picard 반복문을 이용하여 다음 방정식의 근사해를 $y_3$ 까지 구하시오.

$$y^{\prime }+2y-x=0,y(0)=0$$

3. 다음 미분방정식의 일반해를 구하여라.

$$y^{\prime }-y=4$$

4. 다음 1 계 미분방정식의 적분인자 $\mathrm{F}(\mathrm{x})$ 를 구하고 미분방정식을 풀어라.

$$(y+1)dx-(x+1)dy=0$$

5. 다음 함수들은 양의 $x$ 축에서 1 차 독립인가, 또는 1 차 종속인가?
$$\cos x,\sin x,\sin 2x$$

6. $x^2{y}^{\prime \prime }+x(x-2)y^{\prime }-(x-2)y=0$ 의 해, $y_1=x$ 를 구하였다. 또 하나의 해 $y_2$ 를 구하여라.

7. $x^2{y}^{\prime \prime }-4x{y}^{\prime }+6y=\frac{21}{x^4}$ 의 해를 구하여라.

8. 다음 고계미분방정식의 일반해를 구하여라.

$$y^{\prime \prime \prime }+3y^{\prime \prime }+3y^{\prime }+y=8e^x+x+3$$

 

---

Answer

 

1.

$$e^{y^2}=\frac{1}{2}(e^{2x}+1)$$

2.

$$y_1=\frac{1}{2}{x}^2,y_2=\frac{1}{2}{x}^2-\frac{1}{3}{x}^3,y_3=\frac{1}{2}{x}^2-\frac{1}{3}{x}^3+\frac{1}{6}{x}^4$$

3.

$$y=-4+C{e}^x$$

4.

$$F(x)=\frac{1}{(x+1{)}^2},\frac{y+1}{x+1}=C$$

5. Linear independent
6.

$$y=x{e}^{-x}$$

7.

$$y=c_1{x}^2+c_2{x}^3+0.5x^{-4}$$

8.

$$y=(c_1+c_{2}x+c_3{x}^2)e^{-x}+e^x+x$$