Question

Find the value of a so that the differential equation y ' − xy − 6x = 0 has a solution of the form y(x) = a + bex2/2 for any constant

b.

Answer 1

Hello,


`y=a+be^( x^(2) )/(2) \\\\ y'=a'+b*x*e^( x^(2) )/(2) \\\\ y'-xy-6x=0\\\\ a'+b*x*e^( x^(2) )/(2)-b*x*e^( x^(2) )/(2)-ax-6x=0\\\\ a'=x(a+6)\\\\ (a')/(a+6) =x\\\\ ln (a+6)= (x^2)/(2) +c\\\\ a=k*e^( x^(2) )/(2)-6\\\\`

Answer 2

a = -6

Explanation: