Determine the interval for which the following initial value problem has a unique solution?

Question

y''+cos(x)y'+3|x|y=0, y(2)=3, y'(2)=-1
Please explain.

Answer 1

Look up the Existence and Uniqueness Theorem in your book. There will probably be more than one, but focus on the theorem for second order linear equations. It says that for such equations, if the cofficent of y'' is 1, then then a unique solution will exist on any interval (containing the value of x at which the initial conditions are specified) on which the coefficient functions are continuous.

So you have to ask yourself: what is the largest interval containing x=2 on which the functions cosx and 3|x| are continuous?

Answer 2

detect the widely used answer by the choice product rule: (a million + x?)y' + 5x?y = 13x? (a million + x?)(dy / dx) + 5x?y = 13x? d[(a million + x?)y] / dx = 13x? (a million + x?)y = 13 ? x? dx (a million + x?)y = 13x? / 5 + C (x? + a million)y = 13x? / 5 + C y = (13x? / 5 + C) / (x? + a million) y = (13x? + C) / [5(x? + a million)] detect the specific answer by changing for the consistent: whilst x = 0, y = 2 C / 5 = 2 C = 10 y = (13x? + 10) / [5(x? + a million)]