Question

Consider the following system of first order ordinary differential equations: frac{dx_1}{dt} = 2x_1 + x_2 frac{dx_2}{dt} = x_1 + 2x_2 (a) Write the system in the form $x' = Ax$. (b) Find the eigenvalues and eigenvectors of the matrix A. (c) Hence find the general solution to the above system of differential equations. (d) Find the solution if $x_1(0) = 1$, $x_2(0) = 0$. (e) Explain how you know that matrix A is diagonalisable and write matrices P and D such that $D = P^{-1}AP$. (DO NOT find $P^{-1}$ or calculate $P^{-1}AP$.)

          Consider the following system of first order ordinary differential equations:
frac{dx_1}{dt} = 2x_1 + x_2
frac{dx_2}{dt} = x_1 + 2x_2
(a) Write the system in the form $x' = Ax$.
(b) Find the eigenvalues and eigenvectors of the matrix A.
(c) Hence find the general solution to the above system of differential equations.
(d) Find the solution if $x_1(0) = 1$, $x_2(0) = 0$.
(e) Explain how you know that matrix A is diagonalisable and write matrices P and D such that $D = P^{-1}AP$.
(DO NOT find $P^{-1}$ or calculate $P^{-1}AP$.)

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$Consider the following system of first order ordinary differential equations: fracdx1dt = 2x1 + x2 fracdx2dt = x1 + 2x2 (a) Write the system in the form x' = Ax. (b) Find the eigenvalues and eigenvectors of the matrix A. (c) Hence find the general solution to the above system of differential equations. (d) Find the solution if x1(0) = 1, x2(0) = 0. (e) Explain how you know that matrix A is diagonalisable and write matrices P and D such that D = P^-1AP. (DO NOT find P^-1 or calculate P^-1AP.)$

Added by Purificaci-N K.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Anurag Kumar

05/28/2022

Step 1

(a) To write the system in the form x' = Ax, we can rewrite the given equations as follows: x' = [dx1/dt, dx2/dt] = [2x1 + x2, x1 + 2x2] So, the matrix A is: A = [2 1] [1 2] (b) Show more…

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Consider the following system of first order ordinary differential equations: dx1/dt = 2x1 + x2 dx2/dt = x1 + 2x2 (a) Write the system in the form x' = Ax. (b) Find the eigenvalues and eigenvectors of the matrix A. (c) Hence find the general solution to the above system of differential equations. (d) Find the solution if x1(0) = 1, x2(0) = 0. (e) Explain how you know that matrix A is diagonalisable and write matrices P and D such that D = P-1AP. (DO NOT find P-1 or calculate P-1AP.)

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00:01 In this problem, we have to solve the following differential equation, which is d .y is the function of x divided with dx plus 2y is the function of x is equals to 3.

00:12 So, here we can say this is dy divided with dx plus 2y is equal to 3.

00:18 And now, we can say this would be dy divided with dx is equals to 3 minus 2y.

00:26 And now here this would be dy and here this would be 3 minus 2y multiplied with dx.

00:35 Now integrating both hand side, so this would be y is equals to 3x minus this would be 2y multiplied with x.

00:44 So here we have the right answer which is y is equals to this would be 3x minus 2xy and we can say here this would be one more thing.

00:55 Say y plus 2x y is equals to 3x and now we can take y common and 1 plus 2x is here and this is equals to 3x so from here we can say y is equals to 3x divided with 1 plus 2x which is the right answer for this problem and now this is the right answer for part a and now in part we have to find the eigenvalue or we can say eigenvector for the given matrix so, this is 1 and root 2 multiplied with root 2 plus 1.

01:31 So, eigenvalue that means this matrix is multiplied with some values or we can say some matrix and this is giving you 0.

01:39 So we have to find the value of v.

01:41 So here we can say this would be 1 and root 2 multiplied with root 2 1 and this is matrix.

01:49 And here this would be v1, v2 is equals to 0.

01:53 Now we have to do the matrix multiplication.

01:55 So, this would be row into columns...

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