**Problem Analysis**

Let's assume the smaller number as x and the larger number as y.

According to the given information, we have two equations:

1. Difference of square of two numbers is 88:
y^2 - x^2 = 88

2. The larger number is 5 less than twice the smaller number:
y = 2x - 5

We need to solve these equations simultaneously to find the values of x and y.

**Solution**

We will solve the equations using substitution method.

Substitute the value of y from equation 2 into equation 1:

(2x - 5)^2 - x^2 = 88

Expand and simplify the equation:

4x^2 - 20x + 25 - x^2 = 88

Combine like terms:

3x^2 - 20x - 63 = 0

This equation is a quadratic equation. Let's solve it by factoring or using the quadratic formula.

To factorize the equation, we need to find two numbers whose product is -189 (product of the coefficient of x^2 and constant term) and whose sum is -20 (coefficient of x).

After trying different pairs of numbers, we find that the factors are -21 and 9:

3x^2 - 21x + 9x - 63 = 0

Take out common factors from the first two terms and the last two terms:

3x(x - 7) + 9(x - 7) = 0

Now, we have a common binomial factor of (x - 7):

(3x + 9)(x - 7) = 0

Set each factor to zero and solve for x:

3x + 9 = 0 or x - 7 = 0

3x = -9 or x = 7

x = -3 or x = 7

Therefore, we have two possible values for x: -3 and 7.

Now, substitute each value of x back into equation 2 to find the corresponding values of y.

For x = -3:
y = 2(-3) - 5
y = -6 - 5
y = -11

For x = 7:
y = 2(7) - 5
y = 14 - 5
y = 9

So, the two numbers that satisfy the given conditions are -3 and -11, or 7 and 9.

**Conclusion**

The two numbers are either -3 and -11, or 7 and 9.