Question

Given that x²y′+xy=1. 1. Find a solution of the differential equation that satisfies the initial condition y(5)=6. Answer: y=______________ 2. Find a solution of the differential equation that satisfies the initial condition y(6)=5. Answere: y = ____________ .

Answer 1

To find solutions to the given differential equation with initial conditions, use the method of separating variables and solve for the constant of integration.

Step-by-step explanation:

The given differential equation is x²y′+xy=1. To find a solution that satisfies the initial condition y(5)=6, we can use the method of separating variables. Rearranging the equation, we have y′ = (1 - xy) / x². Now, we can separate variables and integrate both sides with respect to y and x, respectively. After solving the integral, we can substitute the initial condition to find the value of the constant of integration and obtain the particular solution:

y = x - 1/x + 7/5

Similarly, to find a solution that satisfies the initial condition y(6)=5, we can follow the same steps and obtain the particular solution:

y = x - 1/x + 2/3