Subtopic: Thermodynamic principles of I.C. engines
Question Statement:
The particular solution to the first order differential equation dy/dx = x − e⁻ˣ for y(0) = −1 is
Options:
(A) 2x²+e⁻ˣ−2
(B) 2x²+e⁻ˣ+½
(C) x²/2−e⁻ˣ+½
(D) x²/2+e⁻ˣ−2
Official Answer Key:
D
Step-by-Step Solution & Explanation
Integrate directly: y = ∫(x−e⁻ˣ)dx = x²/2 + e⁻ˣ + C.
Apply the initial condition y(0)=−1: (0)²/2 + e⁰ + C = −1 ⇒ 0 + 1 + C = −1 ⇒ C = −2.
Hence y = x²/2 + e⁻ˣ − 2, matching option (D). Options (A) and (B) contain the wrong sign on the e⁻ˣ integration term's coefficient (2x² instead of x²/2, from mis-integrating −x instead of x), and (C) has the correct x²/2 term but the wrong sign of e⁻ˣ and the wrong constant.
Practice This in Real CBT Exam Interface
Test yourself with the official TCS-iON palette, virtual calculator, and immediate performance analytics.