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.

$$ y'' + 5 y ' + 4 y = 10 e^{-3x} $$

 1) Solution of homogeneous ODEs.

Let's solve : $ y'' + 5 y ' + 4 y = 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.

$$ y_{p} '' + 5 y_{p} ' + 4 y_{p} = 10 e^{-3x} $$

$$ \therefore K = -5 $$

$$ \therefore y(x) = c_{1} e^{-x} + c_2e^{-4x} - 5^{-3x} $$

 

Problem 2.

Find the solution

$$ 10y'' + 50y' + 57.6y = cos x $$

1) Solution of homogeneous ODE

Let's solve : $ 10y'' + 50y' + 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 : 

$$ y_{p} = K cos x + M sin x $$

And Substitute $ y_{p} $ into nonhomogeneous eq.

 

Problem 3.

Find the solution 

$$ y'' + 3y' + 2y = 12 x^{2} $$

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} $

$$ \therefore K_{2} = 6 \ , \ K_{1} = -6 \ , \ K_{0} = 3 $$

$$ \therefore y(x) = c_{1} e^{-x} + c_{2} e^{-2x} + 6x^{2} - 6x + 3 $$