Maths · Calculus · chapter 9 of 13
Class 12 Maths Chapter 9: Differential Equations MCQs with answers
10 multiple-choice questions on Differential Equations, each with its answer and the reason, written from the NCERT chapter. For the CBSE board and JEE Main, which are set from the same NCERT chapter. Open a question's solution to check your working; then take the free test to find your next chapter.
Question 1
The order of the differential equation (d2ydx2)3 + x (dydx)4 = ex is
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Answer: (d) 2
The derivatives present are d²y/dx² (order 2) and dy/dx (order 1); the order is that of the highest order derivative, 2. Options (b) and (c) read the powers 3 and 4 as orders.
Question 2
The degree of the differential equation (y″)2 + (y′)3 − 7y = 0 is
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Answer: (c) 2
The equation is a polynomial in y′ and y″. The highest order derivative is y″, and its highest power is 2, so the degree is 2. Option (b) takes the highest power of any derivative; (d) adds the two powers.
Question 3
For which of the following differential equations is the degree not defined?
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Answer: (b) y′ + ey′ = x
ey′ is not a polynomial in y′, so (b) is not a polynomial equation in its derivatives and its degree is not defined. In (a) cos y involves y, not a derivative, so (a) is a polynomial in y″ of degree 1; (c) has degree 2 and (d) degree 1.
Question 4
y = a ex + b x ex, where a and b are arbitrary constants, is the general solution of a differential equation. The order of that differential equation is
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Answer: (b) 2
A general solution has as many arbitrary constants as the order of the equation. Here there are two, a and b, so the order is 2.
Question 5
Which of the following is a particular solution of d2ydx2 + 4y = 0?
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Answer: (d) y = 3 cos 2x
For y = 3 cos 2x, y″ = −12 cos 2x, so y″ + 4y = 0, and it has no arbitrary constant: a particular solution. (b) also satisfies the equation but carries the arbitrary constants A, B, so it is the general solution. For (c), y″ + 4y = −36 cos 4x ≠ 0; for (a), y″ + 4y = 8e2x ≠ 0.
Question 6
If y = xm, x > 0, is a solution of x2 y″ − 2x y′ − 4y = 0 and m > 0, then m equals
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Answer: (b) 4
y′ = m xm−1, y″ = m(m − 1) xm−2. Substituting: [m(m − 1) − 2m − 4] xm = 0, so m2 − 3m − 4 = 0, (m − 4)(m + 1) = 0. The positive root is m = 4.
Question 7
The sum of the order and the degree of (d3ydx3)2 − 4(d2ydx2)5 + y = 0 is
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Answer: (c) 5
The highest order derivative is d³y/dx³, so the order is 3; its power is 2, so the degree is 2. The sum is 5. Option (b) uses the power 5 of the second derivative as the degree.
Question 8
Which of the following is not a differential equation?
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Answer: (c) x2 + y2 = 4x
A differential equation must involve a derivative of the dependent variable. x2 + y2 = 4x contains none; (b) is the equation x dy/dx + y = 0 written with differentials.
Question 9
The general solution of dydx = x e−y is
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Answer: (d) ey = x22 + C
Separating, ey dy = x dx. Integrating both sides: ey = x2/2 + C. Option (b) integrates ey wrongly as e−y; (c) drops the factor 1/2.
Question 10
The general solution of dydx = 2xy1 + x2, y > 0, is
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Answer: (c) y = C(1 + x2)
Separating, dy/y = 2x dx/(1 + x2). Integrating: log y = log (1 + x2) + log C, so y = C(1 + x2). Option (a) forgets that the constant multiplies after taking exponentials.
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