Advanced Engineering Math. Problem set 2.7 : 1 ~ 3

Problem 1 ~ 3 의 경우에는 전부 Basic Rule 을 이용하는 단순한 문제다.

Problem 1.

Find the solution.

$$y^{″}+5y^{\prime }+4y=10e^{-3x}$$

 1) Solution of homogeneous ODEs.

Let's solve : $y^{″}+5y^{\prime }+4y=0$

Substitute $y=e^{\lambda x}$ into homogeneous ODEs,

$$({\lambda }^2+5\lambda +4)e^{\lambda }=0$$

By the characteristic equation, D > 0 , The ODE has 2 real roots.

$$\therefore y_h=c_1{e}^{-x}+c_2{e}^{-4x}$$

 2) Solution of Nonhomogeneous ODEs

 To find $y_p$ , Let's apply Basic Rule of method of undetermined coefficients.

Let's assume : 

$$y_p=K{e}^{-3x}$$

Substitute $y_p$ into nonhomogeneous eq.

$$y_p^{″}+5y_p^{\prime }+4y_p=10e^{-3x}$$

$$\therefore K=-5$$

$$\therefore y(x)=c_1{e}^{-x}+c_2{e}^{-4x}-5^{-3x}$$

 

Problem 2.

Find the solution

$$10y^{″}+50y^{\prime }+57.6y=cosx$$

1) Solution of homogeneous ODE

Let's solve : $10y^{″}+50y^{\prime }+57.6y=0$

By the characteristic equation , D > 0 , The ODE has two real roots.

$$\therefore y_h=c_1{e}^{-1.8x}+c_2{e}^{-3.2x}$$

2) Solution of Nonhomogeneous ODE

 Let's apply Basic Rule of method of undetermined coefficients.

Let's assume : 

$$y_p=Kcosx+Msinx$$

And Substitute $y_p$ into nonhomogeneous eq.

 

Problem 3.

Find the solution 

$$y^{″}+3y^{\prime }+2y=12x^2$$

1) Solution of homogeneous ODE

 제차방정식 푸는 문제는 쉽다. 눈으로 봐도 두개의 실근을 갖는 것을 알 수 있다. 따라서, $y_h$ 는,

$$y_h=c_1{e}^{-x}+c_2{e}^{-2x}$$

2) Solution of Nonhomogeneous ODE

Let's apply Basic Rule of method of undetermined coefficients.

 Let's consider the $y_p=K_2{x}^2+K_1{x}^1+K_0$

$$\therefore K_2=6,K_1=-6,K_0=3$$

$$\therefore y(x)=c_1{e}^{-x}+c_2{e}^{-2x}+6x^2-6x+3$$