verify that y=c/x^2 is a general solution of the differential equation y'+(2/x) y=0-6y=0
then find a particular solution of the differential equation that satisfies the side condition y(1)=2
To verify if y = c/x^2 is a general solution of the given differential equation y' + (2/x)y = 0, we need to substitute y = c/x^2 into the differential equation and check if it holds true for any value of c.
1. Substitute y = c/x^2 into the differential equation:
y' + (2/x)y = 0
(c/x^2)' + (2/x)(c/x^2) = 0
(-2c/x^3) + (2c/x^3) = 0
0 = 0
The left side of the equation equals zero, which means the substituted solution is a valid solution for any constant c. Hence, y = c/x^2 is a general solution of the differential equation.
To find a particular solution that satisfies the side condition y(1) = 2, we can substitute x = 1 and y = 2 into the general solution y = c/x^2.
2. Substituting x = 1 and y = 2 into y = c/x^2:
2 = c/1^2
2 = c/1
2 = c
Therefore, c = 2. The particular solution that satisfies the side condition y(1) = 2 is y = 2/x^2.