Solved on Nov 04, 2023

Classify the equilibrium at the origin in X=CXX = CX where C=(911119)C = \begin{pmatrix} 9 & -11 \\ -11 & 9 \end{pmatrix} as a sink, saddle, or source.

STEP 1

Assumptions1. The system of differential equations is given by X=CXX = CX, where XX is a vector of variables, CC is a matrix, and the multiplication is matrix multiplication. . The equilibrium point is at the origin.
3. The matrix CC is given by C=(911119)C = \begin{pmatrix}9 & -11 \\ -11 &9 \end{pmatrix}.

STEP 2

The nature of the equilibrium point at the origin in a system of differential equations can be determined by the eigenvalues of the matrix CC. If all eigenvalues are negative, the equilibrium is a sink. If all eigenvalues are positive, the equilibrium is a source. If the eigenvalues are of mixed sign, the equilibrium is a saddle.

STEP 3

To find the eigenvalues, we need to solve the characteristic equation of the matrix CC, which is given by det(CλI)=0det(C - \lambda I) =0, where $$ is the identity matrix and $\lambda$ is a scalar.

STEP 4

The matrix CλIC - \lambda I is given by CλI=(9λ11119λ)C - \lambda I = \begin{pmatrix}9-\lambda & -11 \\ -11 &9-\lambda \end{pmatrix}.

STEP 5

The determinant of CλIC - \lambda I is given by det(CλI)=(9λ)2(11)2det(C - \lambda I) = (9-\lambda)^2 - (-11)^2.

STEP 6

implify the determinant to get det(CλI)=λ218λ+20det(C - \lambda I) = \lambda^2 -18\lambda +20.

STEP 7

Set the determinant equal to zero and solve for λ\lambda to get the characteristic equation λ218λ+20=0\lambda^2 -18\lambda +20 =0.

STEP 8

The roots of the characteristic equation are the eigenvalues of the matrix CC. To find the roots, we use the quadratic formula λ=b±b24ac2a\lambda = \frac{-b \pm \sqrt{b^2 -4ac}}{2a}, where a=1a =1, b=18b = -18, and c=20c =20.

STEP 9

Substitute the values of aa, bb, and cc into the quadratic formula to get λ=18±(18)24202\lambda = \frac{18 \pm \sqrt{(-18)^2 -4**20}}{2*}.

STEP 10

implify the expression under the square root to get λ=18±324802\lambda = \frac{18 \pm \sqrt{324 -80}}{2}.

STEP 11

Further simplify to get λ=18±244\lambda = \frac{18 \pm \sqrt{244}}{}.

STEP 12

implify the square root to get λ=18±2722\lambda = \frac{18 \pm2*7\sqrt{2}}{2}.

STEP 13

Divide through by2 to get the eigenvalues λ=9±72\lambda =9 \pm7\sqrt{2}.

STEP 14

Since the eigenvalues are both positive, the equilibrium at the origin is a source.

Was this helpful?
banner

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

ContactInfluencer programPolicyTerms
TwitterInstagramFacebookTikTokDiscord