The differential equation, representing the curve y=e^x(a x+b x) where a and b are arbitrary constants, is:

Step-by-step solution

y=e^x(a x+b x) ...(i) aligned d y d x = & e^x(-a x+b x) & +(a x+b x) e^x d y d x = & e^x(-a x+b x)+y aligned Again differentiate w.r.t. to x aligned d^2 y d x^2 = & -e^x(a x+b x) & +(-a x+b x) e^x+ d y d x d^2 y d x^2 = & -y+ d y d x -y+ d y d x [ using (i) and (ii) ] d^2 y d x^2 - & 2 d y d x +2 y=0 aligned is the required differential equation.