Verify by substitution that each given function is a solution of the given differential equation. Primes denote derivatives with respect to x.
1) y'' = 9y ; y1 = e3x , y2 = e-3x
2) y'' - 2y' +2y = 0; y1 = excosx , y2 = exsinx
3) x2y'' + xy' - y = ln x; y1 = x - ln x , y2 = (1/x) - ln xSubstitute y=erx into the given differential equation to determine all values of the constant r for whichy=erx is a solution of the equation.4) 3y''+3y'-4y=0First verify that y(x) satisfies the given differential equation. Then determine a value of the constant C so that y(x) satisfies the given initial condition.
5) y' = y + 1; y(x) = Cex - 1, y(0) = 5
6) xy' - 3y = x3 ; y(x) = x3(C + ln x), y(1) = 17Write a differential equation of the form dy/dx = f(x, y) having the function g as its solution.
7) The line tangent to the graph of g at the point (x, y) intersects the x-axis at the point (x/2 , 0).
8) The time rate of change of a population P is proportional to the square root of P.
9) In a city having a fixed population of P persons, the time rate of change of the number N of those persons who have heard a certain rumor is proportional to the number of those who have not yet heard the rumor.
10) Suppose a population P of rodents satisfies the differential equation dP/dt = kP2. Initially, there are P(0) = 2 rodents, and their number is increasing at the rate of dP/dt = 1 rodent per month when there are P = 10 rodents. How long will it take for this population to grow to a hundred rodents? To a thousand? What's happening here?