Question

The solution of the differential equation \(\frac{dy}{dx} + \frac{y}{x} = x^2\) under the condition that y(1) = 1 is

• A. \(4xy = x^3 + 3\)
• B. \(4xy = x^4 + 3\)
• C. \(4xy = x^3 - 3\)
• D. \(4xy = x^4 - 3\)

Hint:

The integrating factor method is a standard procedure for first-order linear DEs. Remember the three key steps: find P(x) and Q(x), calculate the I.F., and then apply the solution formula \(y \cdot (\text{I.F.}) = \int Q(x) \cdot (\text{I.F.}) \,dx + C\).

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
Solve y'''+3y''+2y'=0.

Step 2: Key Formula (Alternate):
Auxiliary equation: m³+3m²+2m=0. Roots give solution form.

Step 3: Detailed Explanation:
m(m²+3m+2)=0 → m(m+1)(m+2)=0. Roots: 0,-1,-2. y=c₁+c₂e⁻ˣ+c₃e⁻²ˣ.

Step 4: Final Answer:
Solution is y = a+be⁻ˣ+ce⁻²ˣ.