Solve the given initialvalue problem Give the largest interv

Solve the given initial-value problem. Give the largest interval I over which the solution is defined. dy xdx + y = 6x + 1, y(1) = 7 y= 1- (0,00) 0,x)

Solution

Answer.

x dy/dx + y = 6x + 1

dy/dx + (1/x)y = 6 + 1/x

--

Integral factor is given by;

_e1/x dx = e^ln(x) = x

[xy]´ = 6x + 1

so next we solve for;

xy = (6x + 1) dx

xy = 3x² + 1 + C

y = 3x + 1 + C/x

--

For y (1) = 7 we have;

7 = 3(1) + 1 + C/(1)

C = 2

--

Solution is;

y = 3x + 1 + 2/x