CBSE Class 12 Mathematics Differential Equations Important Questions Set A

Multiple Choice Questions

Question. The degree of the differential equation \(x^2 \frac{d^2y}{dx^2} = \left( x \frac{dy}{dx} - y \right)^3\) is 
(a) 1
(b) 2
(c) 3
(d) 6
Answer: (a)

Question. The degree of the differential equation \(\frac{d^2y}{dx^2} + 3\left(\frac{dy}{dx}\right)^2 = x^2 \log \left(\frac{d^2y}{dx^2}\right)\) is 
(a) 1
(b) 2
(c) 3
(d) Not defined
Answer: (d)

Question. The order and degree of differential equation \(\left[ 1 + \left(\frac{dy}{dx}\right)^2 \right]^2 = \frac{d^2y}{dx^2}\) respectively, are
(a) 1, 2
(b) 2, 2
(c) 2, 1
(d) 4, 2
Answer: (c)

Question. The order of the differential equation of all circles of given radius \(a\) is
(a) 1
(b) 2
(c) 3
(d) 4
Answer: (c)

Question. The solution of the differential equation \(2x \cdot \frac{dy}{dx} - y = 3\) represents a family of
(a) straight lines
(b) circles
(c) parabolas
(d) ellipses
Answer: (c)

Question. The integrating factor of the differential equation \(\frac{dy}{dx}(x \log x) + y = 2 \log x\) is 
(a) \(e^x\)
(b) \(\log x\)
(c) \(\log(\log x)\)
(d) \(x\)
Answer: (b)

Question. A solution of the differential equation \(\left(\frac{dy}{dx}\right)^2 - x \frac{dy}{dx} + y = 0\) is 
(a) \(y = 2\)
(b) \(y = 2x\)
(c) \(y = 2x - 4\)
(d) \(y = 2x^2 - 4\)
Answer: (c)

Question. Which of the following is not a homogeneous function of \(x\) and \(y\)?
(a) \(x^2 + 2xy\)
(b) \(2x - y\)
(c) \(\cos^2 \left(\frac{y}{x}\right) + \frac{y}{x}\)
(d) \(\sin x - \cos y\)
Answer: (d)

Question. Solution of the differential equation \(\frac{dx}{x} + \frac{dy}{y} = 0\) is
(a) \(\frac{1}{x} + \frac{1}{y} = c\)
(b) \(\log x \cdot \log y = c\)
(c) \(xy = c\)
(d) \(x + y = c\)
Answer: (c)

Question. The solution of the differential equation \(x \frac{dy}{dx} + 2y = x^2\) is
(a) \(y = \frac{x^2 + C}{4x^2}\)
(b) \(y = \frac{x^2}{4} + C\)
(c) \(y = \frac{x^4 + C}{x^2}\)
(d) \(y = \frac{x^4 + C}{4x^2}\)
Answer: (d)

Question. The degree of the differential equation \(\left( \frac{d^2y}{dx^2} \right)^2 + \left( \frac{dy}{dx} \right)^2 = x \sin \left( \frac{dy}{dx} \right)\) is
(a) 1
(b) 2
(c) 3
(d) Not defined
Answer: (d)

Question. The degree of the differential equation \(\left[ 1 + \left(\frac{dy}{dx}\right)^2 \right]^{3/2} = \frac{d^2y}{dx^2}\) is
(a) 4
(b) \(\frac{3}{2}\)
(c) Not defined
(d) 2
Answer: (d)

Question. The order and degree of a differential equation \(\frac{d^2y}{dx^2} + \left(\frac{dy}{dx}\right)^{\frac{1}{4}} + x^{\frac{1}{5}} = 0\), respectively, are
(a) 2 and not defined
(b) 2 and 2
(c) 2 and 3
(d) 3 and 3
Answer: (a)

Question. If \(y = e^{-x} (A \cos x + B \sin x)\), then it is a solution of
(a) \(\frac{d^2y}{dx^2} + 2 \frac{dy}{dx} = 0\)
(b) \(\frac{d^2y}{dx^2} - 2 \frac{d^2y}{dx^2} + 2y = 0\)
(c) \(\frac{d^2y}{dx^2} + 2 \frac{dy}{dx} + 2y = 0\)
(d) \(\frac{d^2y}{dx^2} + 2y = 0\)
Answer: (c)

Question. Differential equation which has solution of the form \(y = A \cos \alpha x + B \sin \alpha x\), where A and B are arbitrary constants is
(a) \(\frac{d^2y}{dx^2} - \alpha^2 y = 0\)
(b) \(\frac{d^2y}{dx^2} + \alpha^2 y = 0\)
(c) \(\frac{d^2y}{dx^2} + \alpha y = 0\)
(d) \(\frac{d^2y}{dx^2} - \alpha y = 0\)
Answer: (b)

Question. Integrating factor of \(x \frac{dy}{dx} - y = x^4 - 3x\) is
(a) \(x\)
(b) \(\log x\)
(c) \(\frac{1}{x}\)
(d) \(-x\)
Answer: (c)

Question. Solution of \(\frac{dy}{dx} - y = 1\), \(y(0) = 1\) is given by
(a) \(xy = -e^x\)
(b) \(xy = -e^{-x}\)
(c) \(xy = -1\)
(d) \(y = 2e^x - 1\)
Answer: (d)

Question. The number of solution of \(\frac{dy}{dx} = \frac{y + 1}{x - 1}\) when \(y(1) = 2\) is
(a) none
(b) one
(c) two
(d) infinite
Answer: (b)

Question. Which of the following is a second order differential equation?
(a) \((y')^2 + x = y^2\)
(b) \(y'' + y = \sin x\)
(c) \(y''' + (y'')^2 + y = 0\)
(d) \(y' = y^2\)
Answer: (b)

Question. Integrating factor of the differential equation \((1 - x^2) \frac{dy}{dx} - xy = 1\) is
(a) \(-x\)
(b) \(\frac{x}{1 + x^2}\)
(c) \(\sqrt{1 - x^2}\)
(d) \(\frac{1}{2} \log (1 - x^2)\)
Answer: (c)

Fill in the Blanks

Question. The integrating factor of the differential equation \(x \frac{dy}{dx} + 2y = x^2\) is _____________ . 
Answer: \(x^2\)

Question. The number of arbitrary constants in the general solution of a differential equation of order three is _____________ .
Answer: 3

Question. The solution of differential equation \(\cot y dx = x dy\) is _____________ . 
Answer: \(x = C \sec y\)

Question. The degree of the differential equation \(1 + \left( \frac{dy}{dx} \right)^2 = x\) is _____________ . 
Answer: 2

Question. The order of the differential equation \(3 \frac{d^2y}{dx^2} - 5 \left( \frac{dy}{dx} \right)^3 + 2y = 0\) is _____________ .
Answer: 2

Very Short Answer Questions

Question. Find the general solution of the differential equation \(e^{y-x} \frac{dy}{dx} = 1\).
Answer: \(e^{y-x} \frac{dy}{dx} = 1 \Rightarrow \frac{e^y}{e^x} \frac{dy}{dx} = 1 \Rightarrow e^y dy = e^x dx\)
On integrating we have \(\int e^y dy = \int e^x dx \Rightarrow e^y = e^x + C \Rightarrow y = \log (e^x + C)\)

Question. Find the order and degree of differential equation: \(\frac{d^4y}{dx^4} + \sin \left( \frac{d^3y}{dx^3} \right) = 0\). 
Answer: Order is 4 but degree is not defined because given differential equation cannot be written in the form of polynomial in differential co-efficient.

Question. Find the differential equation representing the curve \(y = cx + c^2\). 
Answer: Given \(y = cx + c^2\) ...(i)
\(\Rightarrow \frac{dy}{dx} = c + 0 \Rightarrow \frac{dy}{dx} = c\) [Differentiating with respect to \(x\)]
Putting the value of \(c\) in eq" (i), we get \(y = x \frac{dy}{dx} + \left( \frac{dy}{dx} \right)^2 \Rightarrow \left( \frac{dy}{dx} \right)^2 + x \frac{dy}{dx} - y = 0\)

Question. Find the differential equation representing the curve \(y = e^{-x} + ax + b\), where \(a\) and \(b\) are arbitrary constants. 
Answer: Given curve is \(y = e^{-x} + ax + b\).
\(\Rightarrow \frac{dy}{dx} = -e^{-x} + a\) [Differentiating with respect to \(x\)]
\(\Rightarrow \frac{d^2y}{dx^2} = e^{-x}\) [Differentiating again with respect to \(x\)]

Question. Find the differential equation representing the family of curves \(v = \frac{A}{r} + B\), where \(A\) and \(B\) are arbitrary constants. 
Answer: Given family of curve is \(v = \frac{A}{r} + B\).
\(\frac{dv}{dr} = \frac{-A}{r^2}\) [Differentiating with respect to \(r\)]
\(\frac{d^2v}{dr^2} = \frac{2A}{r^3} \Rightarrow \frac{d^2v}{dr^2} = \frac{2}{r} \cdot \frac{A}{r^2} \Rightarrow \frac{d^2v}{dr^2} = \frac{2}{r} \left( -\frac{dv}{dr} \right)\)
\(\Rightarrow \frac{d^2v}{dr^2} = - \frac{2}{r} \frac{dv}{dr} \Rightarrow r \frac{d^2v}{dr^2} + 2 \frac{dv}{dr} = 0\)

Question. Write the sum of the order and degree of the following differential equation: \(\frac{d}{dx} \left\{ \left( \frac{dy}{dx} \right)^3 \right\} = 0\).
Answer: Given differential equation is \(\frac{d}{dx} \left[ \left( \frac{dy}{dx} \right)^3 \right] = 0 \Rightarrow 3 \left( \frac{dy}{dx} \right)^2 \cdot \frac{d^2y}{dx^2} = 0\).
i.e., order = 2, degree = 1 \(\therefore\) Required sum = 2 + 1 = 3.

Short Answer Questions-I

Question. For a differential equation representing the family of curves \(y = A \sin x\), by eliminating the arbitrary constant. 
Answer: We have, \(y = A \sin x \Rightarrow \frac{y}{\sin x} = A\).
Differentiating with respect to \(x\), we get
\(\frac{\sin x \frac{dy}{dx} - y \cos x}{\sin^2 x} = 0 \Rightarrow \sin x \frac{dy}{dx} - y \cos x = 0\)
\(\Rightarrow \sin x \frac{dy}{dx} = y \cos x \Rightarrow \frac{dy}{dx} = y \cot x\)

Question. Find the differential equation of the family of curves represented by \(y^2 = a(b^2 - x^2)\). 
Answer: We have, \(y^2 = a(b^2 - x^2) = ab^2 - ax^2\).
Differentiating with respect to \(x\), we get
\(2y \frac{dy}{dx} = -2ax \Rightarrow y \frac{dy}{dx} = -ax\) ... (i) \(\Rightarrow \frac{y \frac{dy}{dx}}{x} = -a\) ... (ii)
Again differentiating (i) with respect to \(x\), we get
\(y \frac{d^2y}{dx^2} + \left( \frac{dy}{dx} \right)^2 = -a\).
Using (ii), we get
\(y \frac{d^2y}{dx^2} + \left( \frac{dy}{dx} \right)^2 = \frac{y \frac{dy}{dx}}{x} \Rightarrow xy \frac{d^2y}{dx^2} + x \left( \frac{dy}{dx} \right)^2 - y \frac{dy}{dx} = 0\).

Question. Find the general solution of \(y^2 dx + (x^2 - xy + y^2) dy = 0\). 
Answer: Given, differential equation is \(y^2 dx + (x^2 - xy + y^2) dy = 0\).
\(\Rightarrow y^2 dx = -(x^2 - xy + y^2) dy \Rightarrow \frac{dx}{dy} = -\frac{x^2 - xy + y^2}{y^2}\)
\(\Rightarrow \frac{dx}{dy} = - \left( \frac{x^2}{y^2} - \frac{x}{y} + 1 \right)\) ...(i)
Which is a homogeneous differential equation.
Put \(\frac{x}{y} = v\) or \(x = vy \Rightarrow \frac{dx}{dy} = v + y \frac{dv}{dy}\).
On substituting these values in equation (i), we get
\(v + y \frac{dv}{dy} = -[v^2 - v + 1] \Rightarrow y \frac{dv}{dy} = -v^2 + v - 1 - v = -v^2 - 1\)
\(\Rightarrow y \frac{dv}{dy} = -(v^2 + 1) \Rightarrow \frac{dv}{v^2 + 1} = - \frac{dy}{y}\).
On integrating both sides, we get
\(\tan^{-1}(v) = -\log y + C \Rightarrow \tan^{-1}\left( \frac{x}{y} \right) + \log y = C\)

Question. Solve the differential equation \((y + 3x^2) \frac{dx}{dy} = x\). 
Answer: \((y + 3x^2) dx = xdy \Rightarrow ydx + 3x^2 dx = xdy\)
\(\Rightarrow 3x^2 dx = xdy - ydx \Rightarrow 3dx = \frac{xdy - ydx}{x^2} = d \left( \frac{y}{x} \right)\).
Integrating, we get
\(3x = \frac{y}{x} + C \Rightarrow 3x^2 = y + Cx \Rightarrow y - 3x^2 + Cx = 0\).