Q.30 The solution of the differential equation

Step-by-Step Solution

The given differential equation is a first-order linear differential equation of the form:

dy/dx − y = e^−x

Step 1: Identify the Integrating Factor

For an equation of the form

dy/dx + P(x)y = Q(x),

the integrating factor (IF) is:

IF = e^∫P(x)dx

Here, P(x) = −1

IF = e^−x

Step 2: Multiply the Equation by the Integrating Factor

e^−x dy/dx − e^−x y = e^−2x

The left-hand side becomes:

d/dx (y e^−x)

So,

d/dx (y e^−x) = e^−2x

Step 3: Integrate Both Sides

y e^−x = ∫ e^−2x dx

y e^−x = −1/2 e^−2x + C

Step 4: Solve for y

y = −1/2 e^−x + C e^x

Step 5: Apply Initial Condition y(0) = −1/2

−1/2 = −1/2 (1) + C(1)

C = 0

Final Solution

y = −1/2 e^−x

Correct Answer

✅ Option (C): −1/2 e^−x

Explanation of All Options

Option (A): −1/2 e^−x/2

❌ Incorrect. The exponent −x/2 does not satisfy the structure of the solution derived from the differential equation.

Option (B): −1/2 e^x

❌ Incorrect. This corresponds only to the homogeneous part and does not satisfy the full differential equation or the initial condition.

Option (C): −1/2 e^−x

✅ Correct. This function satisfies:

• the differential equation dy/dx = y + e^−x
• the initial condition y(0) = −1/2

Option (D): −1/2 e^x/2

❌ Incorrect. The exponent x/2 does not appear in the general solution and fails when substituted into the differential equation.

Conclusion

The differential equation was solved using the integrating factor method. After applying the given initial condition,
the unique solution is:

y = −1/2 e^−x

👉 Correct option: (C)