The differential equation of the family of curves $y = e^{x} (A cos x + B s in x)$ , where A and B are arbitrary constants, is

Answer (Detailed Solution Below)View answer ▾

Answer: Option A — $\frac{d ^{2} y}{d x ^{2}} - 2 \frac{d y}{d x} + 2 y = 0$

Detailed Solution

Two arbitrary constants mean a second-order equation. Write $u = e^{x} (B cos x - A s in x)$ .

Differentiating $y$ gives $y' = y + u$ , and differentiating $u$ gives $u' = u - y$ .

$y ″ = y' + u' = y' + u - y = y' + (y' - y) - y = 2 y' - 2 y$ , so $y ″ - 2 y' + 2 y = 0$ .

✓ Check

For

y = e^{x} cos x

y' = e^{x} (cos x - s in x)

and

y ″ = - 2 e^{x} s in x

, so

y ″ - 2 y' + 2 y = - 2 e^{x} s in x - 2 e^{x} cos x + 2 e^{x} s in x + 2 e^{x} cos x = 0

⚠ Common trap

The family

e^{x} (A cos x + B s in x)

corresponds to roots

1 \pm i

, so

r^{2} - 2 r + 2 = 0

. Option (B) has

r^{2} + 2 r - 2 = 0

, whose roots

- 1 \pm 3

are real and cannot produce cosine and sine terms.

Quick tip

Roots $1 \pm i$ : $r^{2} - 2 r + 2 = 0$

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Differential Equations — Preparation

This question tests your understanding of Differential Equations, a key topic in the Quantitative Aptitude section. The detailed solution above uses a step-by-step approach to show the reasoning, and where available, a shortcut that helps you solve similar questions in under 90 seconds during the actual exam.

A differential equation is an equation involving a function and its derivatives. This topic covers the order and degree of a differential equation, forming a differential equation by eliminating arbitrary constants, and solving first-order equations. The main solution methods are variables separable, homogeneous equations (using the substitution y = vx), and linear equations of the form dy/dx + Py = Q, solved with the integrating factor e^(integral of P dx). Exam questions also include application problems on growth and decay, such as population growth, radioactive decay and Newton's law of cooling. Remember that the degree is defined only when the equation is a polynomial in its derivatives. Practise the differential equations questions below, each with a step-by-step solution.

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