Question

Verify by substitution that the given function is a solution of the given differential equation.x2y′′ + xy′ −y = ln x, y = x −ln x

Answer 1

Looks like the differential equation is

x² y'' + x y' - y = ln(x)

and the known solution is

y = x - ln(x)

Take the first derivatives of the solution:

y' = 1 - 1/x

y'' = 1/x²

Substitute y and its derivatives into the DE:

x² (1/x²) + x (1 - 1/x) - (x - ln(x)) = ln(x)

If y is a valid solution, then this equation should reduce to an identity.

1 + x - 1 - x + ln(x) = ln(x)

0 = 0   ✓