Integral Equation

Solve the integral equation: y(x) = 1 + integral from 0 to x of ( y(t)*t^2 dt) (in the integral is y as a function of t times t^2)

Initial value problem

Consider the initial value problem on [1,2]: x^2*y’’ + xy’ – K^2*y = 0, y(1) =1, y’(1)=0 Find the solution y(x,K). Is it a continuous function of K? Can it be differentiated with respect to K? K is a constant

solve system

1. Consider the initial-value problem attached...

linear equation

suppose f=f(x), g=g(x), and I=I(x). Solve the following linear equation to get an implcit solution for I(x): fI' + (f' - g)I = 0 f>0

Find the general solution of the differential equation

Please find the general solution of the differential equation below. It needs to be converted to standard format (s² + bs + c = 0), the roots found, and then plugged into the appropriate general formula that satisfies whether b² + 4c is less than, greater than, or equal to zero. Please show all details in the solution. Thank you 2d²y/dt² + 5dy/dt + 2y = 0