a.
To explain: Whether the statement “The equation
a.
Answer to Problem 1RE
The statement is true.
Explanation of Solution
Given:
The differential equation is
The highest derivative occur in the given equation is two.
Therefore the equation is second order.
Here, it is observed that the differential equation does not contain any product of the function and its derivative and all the power of the derivative are one.
Thus, the nonhomogeneous part of the equation
Therefore, the differential equation is linear and nonhomogeneous.
Thus, the given statement is true.
b.
To explain: Whether the statement “The equation
b.
Answer to Problem 1RE
The given statement is false.
Explanation of Solution
Given:
The differential equation is
The highest derivative occur in the given equation is two. Therefore the equation is second order.
Here, it is observed that the differential equation contains the term
Therefore the equation is nonlinear.
Thus, the nonhomogeneous part of the equation
That is, the differential equation is nonhomogeneous.
Hence, the given statement is false.
c.
To explain: Whether the statement “To solve the equation
c.
Answer to Problem 1RE
The statement is false.
Explanation of Solution
Consider the given differential equation
The above equation of the form
Therefore, it is a Cauchy-Euler equation.
Thus, the trial solution is of the form
That is, cannot solve the equation by using trial solution
Hence, the given statement is false.
d.
To explain: Whether the statement “To find the particular solution of the given differential equation, using the trial solution as
d.
Answer to Problem 1RE
The statement is false.
Explanation of Solution
Consider the given differential equation
Here, the homogeneous part of the equation is
The characteristic polynomial of the homogeneous part of the equation is,
Now, the general solution of the homogeneous part of the equation is
Therefore, the function
Thus, the trial solution is
Hence, the statement is false.
e.
To explain: Whether the statement “The function
e.
Answer to Problem 1RE
The statement is true.
Explanation of Solution
Consider the general form of the second order linear homogeneous equation,
In the above equation, there is no separate term of function of t and each additive terms associate with
Therefore, each additive term will be zero.
Thus,
Hence, the given statement is true.
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