Question

The functions y=x2+cx2 are all solutions of equation: xy′+2y=4x2, (x>0). Find the constant c which produces a solution which also satisfies the initial condition y(10)=2.

Answer 1

Answer: c = -0.98

Explanation: Differential equation of first order is an equation involving differentiable function and its derivative.

For the differential equation:
`xy' +2y=4x^(2)`, it given a solution function:

`y=x^(2)+cx^(2)`

Factorating the function:
`y=x^(2)(1+c)`

So, to determine constant c, use the initial condition:

When x = 10, y = 2:

`2=10^(2)(1+c)`

`1+c=(2)/(100)`

`1+c=0.02`

c = -0.98

Constant c that satisfies the initial condition is c = -0.98