To find the general solution of the differential equation given by 8y y” = 0, we start by rewriting the equation in a more manageable form.
First, we can factor the equation:
8y y” = 0 implies either y = 0 or y” = 0.
1. If y = 0, this is a particular solution to the equation.
2. If y” = 0, we have a second-order differential equation that suggests that the function y can be expressed as a linear function of x.
Integrating y” = 0 gives:
y’ = C_1 (where C_1 is a constant).
Integrating again:
y = C_1 x + C_2 (where C_2 is another constant).
Therefore, the general solution of the differential equation is:
y = C_1 x + C_2 for constants C_1 and C_2.
In summary, the general solution combines the linear terms derived from the integration of the second derivative set to zero, along with the specific solution where y = 0.