Find the general solution of the given differential equation. x dy dx − y = x2 sin(x) give the largest interval over which the general solution is defined. (think about the implications of any singular points. enter your answer using interval notation.
Divide both sides by . In doing so, we force any possible solutions to exist on either or (the "positive" interval in such a situation is usually taken over the "negative" one) because cannot be 0 in order for us to do this.
Condense the left side as the derivative of a product, then integrate both sides and solve for :
Note that in order to do this division, we cannot allow . This means the largest interval on which a solution can exist is either or .
If is a solution to the ODE, then any term that vanishes as (or , depending on which interval above is used) is a transient term.
Solve the ODE:
As , will oscillate between -1 and 1, so will oscillate between and , so the limit of as does not exists. There are no transient terms.
Divide both sides by - note that this means we can't have :
Then the left side reduces to the derivative of a product,
This solution is continuous everywhere, but accounting for the singular point , the largest interval over which it is defined would be or .