The discussion centers on the solution to the ordinary differential equation (ODE) \( x \frac{dy}{dx} - y = x^2 \sin{x} \), with the solution given as \( y = cx - x \cos{x} \) over the interval \( (0, \infty) \). The restriction to positive values for \( x \) is due to the requirement for continuity in the solution, as the ODE cannot be expressed in the required form at \( x = 0 \). The participants explore the implications of initial conditions and the necessity of avoiding discontinuities in the solution's interval.
PREREQUISITESMathematics students, educators, and professionals working with differential equations, particularly those interested in the behavior of solutions and their intervals of validity.
MarkFL said:I can see why $x=0$ is eliminated in order to write the ODE in the form:
$$\frac{d}{dx}\left(\frac{y}{x}\right)=\sin(x)$$
But I don't know why they have restricted $x$ only to positive values. :D
find_the_fun said:Isn't it because interval has to be continuous i.e. can't have a break at 0?