Find the general solution of the given differential equation.
x dy/dx − y = x^2 sin x
I have solved to:
y = -x cos(x) + cx
The second part of the question is to give the largest interval over which the general solution is defined.
I would think it would be - infinity to + infinity but webassign says that is wrong. How do you determine the domain?
to solve it, did you divide by x?
To solve for y, yes.
To determine the domain of the general solution, we need to consider any restrictions or limitations that may exist in the differential equation.
The given differential equation is:
x(dy/dx) - y = x^2 sin(x)
To find the domain of the general solution, we need to consider any points at which the equation is not defined. In this case, we observe that the coefficient of dy/dx is x, which means that the equation is not defined at x = 0.
To find the largest interval over which the general solution is defined, we need to consider the range of values for x where the equation is valid. Since x = 0 is excluded from the domain, the interval would be the set of all real numbers except x = 0.
Therefore, the largest interval over which the general solution is defined is (-∞, 0) U (0, ∞).
To determine the largest interval over which the general solution of the differential equation is valid, we need to consider the domain restrictions imposed by the given equation.
Given the differential equation:
x(dy/dx) - y = x^2 sin(x)
First, we need to analyze the terms involved. The expression "x dy/dx" suggests that the function y must be differentiable and defined on an interval where dy/dx exists. Additionally, the term "x^2 sin(x)" implies that sin(x) must be defined for the given x values.
To find the domain of the general solution, we can examine the possible restrictions of the terms involved:
1. x(dy/dx): As long as the function y is differentiable, there are no explicit restrictions on x(dy/dx).
2. y: Since y = -x cos(x) + cx is the general solution obtained, it is defined for all real values of x.
3. x^2 sin(x): The sine function sin(x) is defined for all real numbers, so there are no restrictions on x^2 sin(x).
Now, to determine the largest interval over which the general solution is defined, we need to consider any implicit restrictions that might arise.
In this particular differential equation, there are no specific restrictions imposed by the equation itself. Thus, the general solution y = -x cos(x) + cx is defined on the entire real line, which means that the largest interval over which the general solution is valid is (-∞, +∞).
Therefore, the correct answer is indeed (-∞, +∞). It seems there might be a mistake in the feedback provided by webassign.