Subtopic: Linear and non-linear first order Ordinary Differential Equations (ODE)
Question Statement:
The solution of $x\frac{dy}{dx} = x^2 + 3y$ for $x > 0$ and $y(1) = 2$ is:
Options:
(A) y = x³ + 3x²
(B) y = x² + 3x³
(C) y = −x² + 3x³
(D) y = −x² − 3x³
Official Answer Key:
B
Step-by-Step Solution & Explanation
Rewrite as dy/dx − (3/x)y = x, a linear first-order ODE. Integrating factor IF = e^(−∫3/x dx) = x⁻³. Then d/dx(y·x⁻³) = x·x⁻³ = x⁻². Integrating: y·x⁻³ = −x⁻¹ + C ⇒ y = −x² + Cx³. Apply y(1)=2: 2 = −1 + C ⇒ C = 3. So y = −x² + 3x³. Checking the options, this matches option (C) algebraically (y = −x² + 3x³); however the official key records option (B), y = x² + 3x³, which would follow if the sign of the −3/x term in the integrating factor step is taken as +3/x (i.e., IF = x⁻³ misapplied as x³). Official key: (B). Students should re-derive carefully — the mathematically consistent solution satisfying both the ODE and y(1)=2 is y = −x² + 3x³ (option C); this discrepancy with the official key is noted here for transparency.
Section 2 — Farm Machinery
9 question(s) in this section: Q.3, Q.4, Q.16, Q.31, Q.32, Q.33, Q.34, Q.50, Q.51
Practice This in Real CBT Exam Interface
Test yourself with the official TCS-iON palette, virtual calculator, and immediate performance analytics.