The general solution of the equation,
Answer to Problem 1P
The general solution of
Explanation of Solution
The given system of equations is,
It is noted that, the given system is a non-homogeneous system.
Consider the homogeneous part. That is,
This is of the form,
The eigenvalues r and the eigenvectors
The roots of the above characteristic equation is obtained as follows.
Thus, the solutions are,
Set
On solving,
Let
Then, the eigen vector formed is,
Thus, the first solution of the homogeneous solution is,
Set
On solving,
Let
Then, the eigen vector formed is,
Thus, the second solution of the homogeneous solution is,
Therefore, the matrix T of eigen vectors is,
Then, the inverse of T is,
It is noted from the given equation that,
Now consider the equation,
Substitute the eigen values and inverse of T in the equation (1) as follows.
Thus, the following two linear differential equations are obtained.
Consider the equation (2). The integral factor is,
Multiply the obtained integral factor throughout the equation (2) and
Similarly, multiply the obtained integral factor
The solution is given by the equation,
Substitute the obtained solutions in the equation (6) as follows.
Therefore, the general solution of
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Chapter 7 Solutions
Elementary Differential Equations
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