Show that the following equation yx expintegralxx0 pt dt is

Show that the following equation:

y(x) - exp{integral^x_x0 p(t) dt} is a solution to the differential equation y\" + ax(x)tf + a2(x)y = 0 provided p(x) is a solution to the Riccati equation J = - y\' - ai(x)tj -a^(x).

Solution

Differentiating , we get

y\'= y p(x)

y\'\'= y\' p(x) + y p\'(x)

    = y p(x)² + yp\'(x)

So y\'\'+ a₁ y\' +a₂ y = y[p(x)² +p\'(x)+ a₁ p+a₂] =y 0= 0 if p(x) satisfies the Ricatti equation.