Jawaharlal Nehru Technological University Hyderabad
Jawaharlal Nehru Technological University Hyderabad B.Tech I Year ORDINARY DIFFERENTIAL EQUATIONS AND VECTOR CALCULUS Previous Question Paper (September - 2023 - R22)
Note: This question paper contains two parts A and B. Part-A for 10 marks, Part - B for 50 marks. Part-A is a compulsory question. Part-B consists of ten questions (numbered from 2 to 11) carrying 10 marks each. From each unit, there are two questions and the student should answer one of them. Hence, the student should answer five questions from Part-B.
Part A
Define an exact equation.
Answer
A first-order differential equation of the form M(x, y)dx + N(x, y)dy = 0 is said to be exact if there exists a function f(x, y) such that df = Mdx + Ndy, which is equivalent to the condition ∂M/∂y = ∂N/∂x.
Write the standard form of a linear differential equation, in x.
Answer
The standard form of a linear differential equation in y is dy/dx + P(x)y = Q(x), where P and Q are functions of x only.
Find general solution of the differential equation y'' + y' - 2y = 0.
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Find the PI of (D2 + 4D + 13)y = 2e-x.
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State first shifting theorem of Laplace transforms.
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State convolution theorem of Laplace transforms.
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Define a scalar point function.
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Define the divergence of a vector point function.
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Define work done by force F in moving a particle from A to B along the curve C.
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Define a volume integral.
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Part B
Solve (ex + 1)cos x dx + eysin x dy = 0.
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If the air is maintained at 20°C and the temperature of the body cools from 100°C to 80°C in 10 minutes, find the temperature of the body after 20 minutes and when the temperature will be 40°C?
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Solve (1 + y2)dx = (tan-1y - x)dy.
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Prove that the system of confocal and coaxial parabolas y2 = 4a(x + a) is self-orthogonal.
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Solve (D2 + 1)y = sin x sin 2x.
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Solve (D2 + 4)y = tan 2x by method of variation parameters.
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An LCR circuit with battery emf E sin pt is tuned to resonance so that p2 = 1/(LC). If initially the current i and the charge Q be zero, then show that for small values of R/L, the current in the circuit at time t is given by (E/2L)t sin pt.
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Find the Laplace transform of f(t) = e-2t + 3 sin 2t - 2 cos 3t + t + 100.
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Use convolution theorem to find the inverse Laplace transform of 1/((s+1)(s+2)).
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Find the Laplace transform of Dirac delta function.
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Using Laplace transform, solve the differential equation (D2 - 3D - 4)y = 2e-t with y(0) = 1 = y'(0).
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Find the directional derivative of Φ = 2xy + z2 at (1, -1, 3) in the direction of a vector i + 2j + 3k.
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If r = xi + yj + zk, prove that div r = 3 and curl r = 0.
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Find the values of a, b, and c so that the vector F = (x + y + az)i + (bx + 2y - z)j + (-x + cy + 2z)k be irrotational.
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Prove that Curl (F × G) = (G.∇)F - G div F - (F.∇)G + F div G.
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If F = 3x2yi + (x3 - 2yz2)j + (3z2 - 2y2z)k, then:
Prove that F is conservative.
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Find its scalar potential Φ.
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Find the work done in moving a particle under this force field from (2, 1, 1) to (2, 0, 1).
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Verify Green’s theorem in plane for ∫(3x2 - 8y2)dx + (4y - 6xy)dy, where C is the region bounded by the parabolas y2 = x and x2 = y.
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