Problem Statement

Solve the second-order homogeneous differential equation

d²y/dx² − y = 0

subject to the initial conditions:

• y(0) = 1
• dy/dx (0) = 3

and determine the value of y at x = 2.

Step 1: Characteristic Equation

Assume a solution of the form:

y = e^rx

Substituting into the differential equation gives:

r² − 1 = 0

Solving:

r = ±1

Step 2: General Solution

The general solution is:

y(x) = c₁e^x + c₂e^−x

This can also be written using hyperbolic functions:

y(x) = A cosh(x) + B sinh(x)

Step 3: Apply Initial Conditions

Condition 1: y(0) = 1

c₁ + c₂ = 1

Condition 2: y′(0) = 3

First derivative:

y′(x) = c₁e^x − c₂e^−x

At x = 0:

c₁ − c₂ = 3

Solving the Two Equations

Adding:

2c₁ = 4 → c₁ = 2

Therefore:

c₂ = −1

Final Form of the Solution

y(x) = 2e^x − e^−x

or equivalently:

y(x) = 2 cosh(x) + sinh(x)

Step 4: Evaluate y(2)

y(2) = 2e² − e⁻²

Using numerical values:

• e² ≈ 7.389
• e⁻² ≈ 0.135

y(2) = 2(7.389) − 0.135

= 14.778 − 0.135

= 14.64

Final Answer

y(2) ≈ 14.64

Exam Notes

• This is a standard linear homogeneous ODE with constant coefficients
• Always apply initial conditions after finding the general solution
• Hyperbolic and exponential forms are mathematically equivalent