Rajiv Gandhi Proudyogiki Vishwavidyalaya, Bhopal
New Scheme Based On AICTE Flexible Curricula
Common to All Disciplines | II Semester
3L-1T-0P 4 Credits
Module 1: Ordinary Differential Equations I (6 hours)
Differential Equations of First Order and First Degree (Leibnitz linear, Bernoulli’s, Exact), Differential Equations of First Order and Higher Degree, Higher order differential equations with constants coefficients, Homogeneous Linear Differential equations, Simultaneous Differential Equations.
Previous Years questions appears in RGPV exam.
Q.1) Solve \( \cos x \frac{dy}{dx} = y(\sin x - y) \) using Bernoulli's. (Nov-2022)
Q.2) Solve the Linear differential equation \( \sin 2x \frac{dy}{dx} - y = \tan x \). (Nov-2022)
Q.3) Solve \( (r + \sin\theta - \cos\theta)dr + r(\sin\theta + \cos\theta)d\theta = 0 \). (Nov-2022)
Q.4) Solve the differential equation. \( (D^3 - 7D^2 + 14D - 8)y = e^x \cos 2x \). (Nov-2022)
Q.5) Solve: \( \frac{dy}{dx} = \cos(x+y) + \sin(x+y) \). (June-2023)
Q.6) Solve: \( (1+y^2)dx = (\tan^{-1}y-x)dy \). (June-2023, Dec-2024)
Q.7) Solve: \( \frac{d^2y}{dx^2} + \frac{dy}{dx} = (1+e^x)^{-1} \). (June-2023)
Q.8) Solve: \( \frac{dx}{dt} - y = e^t \), \( \frac{dy}{dt} + x = \sin t \); with initial conditions \( x(0)=1, y(0)=0 \). (June-2023)
Q.9) Solve \( (1+y^2)dx = (\tan^{-1}y - x)dy \) using Leibnitz linear method. (Dec-2023)
Q.10) Solve \( (e^y+1)\cos x dx + e^y \sin x dy = 0 \). (Dec-2023)
Q.11) Solve \( (D^2 - 4D + 3)y = \cos 2x \). (Dec-2023)
Q.12) Solve \( x\frac{dy}{dx} + y = x^3y^6 \) using Bernoulli's. (June-2024)
Q.13) Solve the differential equation \( (xe^y+2y)\frac{dy}{dx} + y e^y = 0 \) using Exact method. (June-2024)
Q.14) Solve \( (D^2-6D+13)y = 8e^{3x}\sin 2x \). (June-2024)
Q.15) Solve \( (D^2+3D+2)y = \sin 3x \). (Dec-2024)
Q.16) Solve the simultaneous equations \( \frac{dx}{dt} - 7x + y = 0 \) and \( \frac{dy}{dt} - 2x - 5y = 0 \). (Dec-2024)
Q.17) Solve \( (e^x+1)\cos x\ dx + e^x\sin x\ dy = 0 \). (June-2025)
Q.18) Solve \( (D^2-5D+6)y = 4e^x+5 \). (June-2025)
Expected Sample Questions for Dec-2025 Exam (Based on Syllabus Analysis)
Q.1) Solve the Bernoulli's equation: \( \frac{dy}{dx} + \frac{y}{x} = y^2 \). (Predicted)
Q.2) Solve the simultaneous differential equations: \( \frac{dx}{dt} + 2y = -\sin t \), \( \frac{dy}{dt} - 2x = \cos t \). (Predicted)
Q.3) Solve the differential equation with constant coefficients: \( (D^3 - 1)y = 0 \). (Predicted)
Q.4) Solve the exact differential equation: \( (y^2 e^{xy^2} + 4x^3)dx + (2xy e^{xy^2} - 3y^2)dy = 0 \). (Predicted)
Q.5) Solve the homogeneous linear differential equation: \( x^2 \frac{d^2y}{dx^2} - 3x \frac{dy}{dx} + 4y = 2x^2 \). (Predicted)
Module 2: Ordinary differential Equations II (8 hours)
Second order linear differential equations with variable coefficients, Method of variation of parameters, Power series solutions; Legendre polynomials, Bessel functions of the first kind and their properties.
Previous Years questions appears in RGPV exam.
Q.1) Solve \( (D^2 + 4)y = \tan 2x \) by using method of variation of parameters. (Nov-2022)
Q.2) Solve the differential equation \( x(1-x)y'' + 2(1-2x)y' - 2y = 0 \) using Frobenius method. (June-2023)
Q.3) Prove that \( J_{\frac{1}{2}}(x) = \sqrt{\frac{2}{\pi x}}\sin x \). (June-2023, Dec-2023)
Q.4) Solve \( (D^2+9)y = \tan 3x \) by using method of variation of parameters. (Dec-2023)
Q.5) Show that \( \frac{d}{dx}[x^n J_n(x)] = x^n J_{n-1}(x) \). (June-2024)
Q.6) Solve \( (D^2+1)y = x \sin x \) using variation of parameters. (June-2024)
Q.7) Solve by the method of variation of parameter \( (D^2+1)y=x \). (Dec-2024)
Q.8) Solve \( (1+x)^2 \frac{d^2y}{dx^2} + (1+x)\frac{dy}{dx} + y = \cos \log(1+x) \). (Dec-2024)
Q.9) Show that \( J_n(-x) = (-1)^n J_n(x) \) when n is positive or negative integer. (Dec-2024)
Q.10) Solve \( x^2\frac{d^2y}{dx^2} + 5x\frac{dy}{dx} + 4y = x\log x \). (June-2025)
Q.11) Solve in series Legendre's differential equation \( (1-x^2)\frac{d^2y}{dx^2} - 2x\frac{dy}{dx} + 2y = 0 \). (June-2025)
Q.12) Solve \( (D^2+a^2)y = \tan ax \) by using method of variation of parameters. (June-2025)
Expected Sample Questions for Dec-2025 Exam (Based on Syllabus Analysis)
Q.1) Solve by method of variation of parameters: \( \frac{d^2y}{dx^2} + y = \sec x \). (Predicted)
Q.2) Solve the Cauchy-Euler equation: \( x^2 \frac{d^2y}{dx^2} - 2x \frac{dy}{dx} - 4y = x^4 \). (Predicted)
Q.3) Solve in series: \( (1-x^2)y'' - 2xy' + n(n+1)y = 0 \). (Predicted)
Q.4) Prove the recurrence relation: \( 2J_n'(x) = J_{n-1}(x) - J_{n+1}(x) \). (Predicted)
Q.5) Express \( f(x) = x^3 + 2x^2 - 4x + 5 \) in terms of Legendre polynomials. (Predicted)
Module 3: Partial Differential Equations (8 hours)
Formulation of Partial Differential equations, Linear and Non-Linear Partial Differential Equations, Homogeneous Linear Partial Differential Equations with Constants Coefficients.
Previous Years questions appears in RGPV exam.
Q.1) Solve \( x^2p^2 + y^2q^2 = z^2 \). (Nov-2022)
Q.2) Solve \( (D^2 - 4DD' + 4D'^2)Z = \cos(x-2y) \). (Nov-2022)
Q.3) Solve by Charpit's method, the P.D.E \( (p^2+q^2)y = qz \). (June-2023, Dec-2023)
Q.4) Solve: \( (D^2 - 6DD' + 9D'^2)z = 12x^2 + 36xy \). (June-2023)
Q.5) Solve the partial differential equation \( (x-y)p + (x+y)q = 2xz \). (Dec-2023)
Q.6) Solve \( (D^2+4DD'-5D'^2)Z = \sin(2x+3y) \). (Dec-2023)
Q.7) Form the partial differential equation (By eliminating the arbitrary functions) from \( Z=(x+y)\phi(x^2-y^2) \). (June-2024)
Q.8) Solve \( (D^2-DD'-6D'^2)Z = xy \). (June-2024)
Q.9) Solve the partial differential equation \( yp-xp=z \). (June-2024)
Q.10) Solve by Charpit's method, \( px+qy=pq \). (Dec-2024)
Q.11) Solve the Partial differential equation \( (D^3 - 4D^2D' + 4DD'^2)Z = \cos(2x+y) \). (Dec-2024)
Q.12) Construct a partial differential equation from the relation \( f(x^2+y^2+z^2, z^2-2xy) = 0 \). (Dec-2024)
Q.13) Eliminate the arbitrary function f from the relation \( z = y^2 + 2f(\frac{1}{x} + \log y) \). (June-2025)
Q.14) Solve the partial differential equation \( \frac{\partial^2 z}{\partial x^2} - \frac{\partial^2 z}{\partial y^2} = x^2y \). (June-2025)
Q.15) Solve \( (y+z)p + (x+z)q = x+y \). (June-2025)
Expected Sample Questions for Dec-2025 Exam (Based on Syllabus Analysis)
Q.1) Solve using Charpit's method: \( px + qy = pq \). (Predicted)
Q.2) Solve the homogeneous linear PDE: \( (D^2 - DD' - 2D'^2)z = (y-1)e^x \). (Predicted)
Q.3) Form the PDE by eliminating arbitrary functions from \( z = f(x+ay) + g(x-ay) \). (Predicted)
Q.4) Solve the linear PDE using Lagrange's method: \( x(y^2-z^2)p + y(z^2-x^2)q = z(x^2-y^2) \). (Predicted)
Q.5) Solve: \( (D^2 - D'^2)z = \cos(x+y) \). (Predicted)
Module 4: Functions of Complex Variable (8 hours)
Functions of Complex Variables: Analytic Functions, Harmonic Conjugate, Cauchy-Riemann Equations (without proof), Line Integral, Cauchy-Goursat theorem (without proof), Cauchy Integral formula (without proof), Singular Points, Poles & Residues, Residue Theorem, Application of Residues theorem for Evaluation of Real Integral (Unit Circle).
Previous Years questions appears in RGPV exam.
Q.1) Show that \( f(Z) = Z\bar{Z} \) is differentiable but not analytic at origin. (Nov-2022)
Q.2) Show that \( u(x,y) = e^{-2x} \sin 2y \) is harmonic and determine it's Harmonic conjugate. (Nov-2022)
Q.3) By Residue theorem, Evaluate \( \oint_C \frac{\tan z}{z^2-1} dz \), where C:|Z|=2. (Nov-2022)
Q.4) Using Cauchy integral theorem, to evaluate the integral \( \oint_C \frac{e^{2z}}{(z-1)^2(z-3)} dz \), where C is the circle |Z|=2. (Nov-2022)
Q.5) Prove that an analytic function with constant modulus is constant. (June-2023)
Q.6) Use Cauchy Integral formula to solve \( \oint_C \frac{\sin \pi z^2 + \cos \pi z^2}{(z-1)(z-2)}dz \) where C is the circle |z|=3. (June-2023)
Q.7) Using complex integration method, solve: \( \int_0^{2\pi} \frac{\cos 4\theta}{5+4\cos\theta}d\theta \). (June-2023)
Q.8) Solve: \( \int_0^{1+i} (x-y+ix^2)dz \) along the real axis from \( z=0 \) to \( z=1 \) and then along a line parallel to imaginary axis from \( z=1 \) to \( z=1+i \). (June-2023)
Q.9) Determine p so that the function \( f(z) = \frac{1}{2}\log(x^2+y^2) + i \tan^{-1}(\frac{px}{y}) \) is analytic function. (Dec-2023)
Q.10) Show that the function \( u(x,y) = e^x \cos y \) is Harmonic. Determine it's Harmonic conjugate. (Dec-2023)
Q.11) Find the residue of \( \frac{Ze^z}{(Z-1)^3} \) at it's pole. (Dec-2023)
Q.12) Write short note on: Cauchy Riemann equations. (Dec-2023)
Q.13) Show that \( u = e^{-x}(x \sin y - y \cos y) \) is Harmonic. (June-2024, Dec-2024)
Q.14) Evaluate \( \int_{(0,0)}^{(1,1)} (3x^2+4xy+ix^2)dz \) along \( y=x^2 \). (June-2024, June-2025)
Q.15) Find the Poles and Residues at each pole of \( f(z) = \frac{\sin^2 z}{(z-\frac{\pi}{6})^2} \). (June-2024)
Q.16) Write short note on: Cauchy's integral formula. (June-2024)
Q.17) Determine P such that the function \( f(z) = \frac{1}{2}\log(x^2+y^2) + i \tan^{-1}(\frac{px}{y}) \) be an analytic function. (Dec-2024)
Q.18) Evaluate using Cauchy's theorem \( \oint_C \frac{z^2e^{-z}}{(z-1)^2}dz \) where c is \( |z-1|=\frac{1}{2} \). (Dec-2024)
Q.19) Find the poles and residues at each pole of \( \frac{e^z}{z^2+1} \). (Dec-2024)
Q.20) Find all values of K such that \( f(z) = e^x(\cos ky + i\sin ky) \) is analytic. (June-2025)
Q.21) If \( f(z) = \frac{1}{(z-1)(z-2)^2} \) find residue of all poles. (June-2025)
Expected Sample Questions for Dec-2025 Exam (Based on Syllabus Analysis)
Q.1) Show that \( u = x^3 - 3xy^2 \) is harmonic and find its harmonic conjugate. (Predicted)
Q.2) Evaluate \( \int_C \frac{e^{2z}}{(z+1)^4} dz \) where \( C \) is \( |z|=3 \) using Cauchy Integral Formula. (Predicted)
Q.3) Evaluate \( \int_0^{2\pi} \frac{d\theta}{5+3\cos\theta} \) using Residue theorem. (Predicted)
Q.4) Determine \( p \) such that \( f(z) \) is analytic. (Predicted)
Q.5) Find the residue of \( f(z) = \frac{z^2}{(z-1)(z-2)^2} \) at its poles. (Predicted)
Module 5: Vector Calculus (10 hours)
Differentiation of Vectors, Scalar and vector point function, Gradient, Geometrical meaning of gradient, Directional Derivative, Divergence and Curl, Line Integral, Surface Integral and Volume Integral, Gauss Divergence, Stokes and Green theorems.
Previous Years questions appears in RGPV exam.
Q.1) Show that \( \frac{\vec{r}}{r^3} \) is solenoidal. (Nov-2022)
Q.2) Show that the vector \( (x^2 - yz)\hat{i} + (y^2 - zx)\hat{j} + (z^2 - xy)\hat{k} \) is irrotational. Find it's scalar potential. (Nov-2022)
Q.3) Verify Green's theorem for \( \oint_C [(3x^2 - 8y^2) dx + (4y - 6xy) dy] \). Where C is the region bounded by \( x=0, y=0 \) and \( x+y=1 \). (Nov-2022)
Q.4) Prove that: \( \nabla^2 f(r) = f''(r) + \frac{2}{r}f'(r) \). (June-2023)
Q.5) Find the directional derivative of \( f(x,y,z) = e^{2x} \cos yz \) at \( (0,0,0) \) in the direction of the tangent to the curve \( x=a \sin t, y=a \cos t, z=at \) at \( t=\frac{\pi}{4} \). (June-2023)
Q.6) Using Green's theorem, find the area of the region in the first quadrant bounded by the curve \( y=x, y=\frac{1}{x}, y=\frac{x}{4} \). (June-2023)
Q.7) Verify Gauss divergence theorem for \( \vec{F} = x^3\hat{i} + y^3\hat{j} + z^3\hat{k} \) taken over the cube bounded by \( x=0, x=a, y=0, y=a, z=0, z=a \). (Dec-2023)
Q.8) Prove that \( curl(r^n \vec{r}) = \vec{0} \). (Dec-2023)
Q.9) Write short note on: Stokes theorem. (Dec-2023)
Q.10) Verify Green's theorem in the plane for \( \oint_C [(x^2-xy^3)dx+(y^2-2xy)dy] \) where C is a square with vertices (0,0), (2,0), (2,2), (0,2). (June-2024)
Q.11) Find the directional derivative of \( f(x,y,z) = xy^2+yz^3 \) at point (2, -1, 1) in the direction of the vector \( \hat{i}+2\hat{j}+2\hat{k} \). (June-2024)
Q.12) Write short note on: Solenoidal and Irrotational. (June-2024)
Q.13) Find the directional derivative of \( \emptyset = x^2yz + 4xz^2 \) at \( (1, -2, -1) \) in the direction of \( 2\hat{i} - \hat{j} - 2\hat{k} \). (Dec-2024)
Q.14) Verify Green's theorem for \( \oint_C [(xy+y^2)dx + x^2dy] \) where C is the boundary by \( y=x \) and \( y=x^2 \). (Dec-2024)
Q.15) If \( \vec{r} \) is the position vector of any point in space, then prove that \( r^n\vec{r} \) is irrotational. (June-2025)
Q.16) Find the workdone by the force \( \vec{F} = z\hat{i} + x\hat{j} + y\hat{k} \), when it moves a particle along the arc of the curve \( \vec{r} = \cos t\hat{i} + \sin t\hat{j} - t\hat{k} \) from \( t=0 \) to \( t=2\pi \). (June-2025)
Q.17) Verify stokes theorem for \( \vec{F}=(x^2-y^2)\hat{i} + 2xy\hat{j} \) over the box bounded by the planes \( x=0, x=a, y=0, y=b \). (June-2025)
Expected Sample Questions for Dec-2025 Exam (Based on Syllabus Analysis)
Q.1) Find the directional derivative of \( \phi = xy + yz + zx \) in the direction of vector \( \vec{a} \). (Predicted)
Q.2) Verify Gauss Divergence theorem for \( \vec{F} = 4x\hat{i} - 2y^2\hat{j} + z^2\hat{k} \) over the cylinder. (Predicted)
Q.3) Verify Stokes theorem for \( \vec{F} = (x^2+y^2)\hat{i} - 2xy\hat{j} \) taken around the rectangle. (Predicted)
Q.4) Show that \( \vec{F} = (y^2+2xz^2)\hat{i} + (2xy-z)\hat{j} + (2x^2z-y)\hat{k} \) is irrotational and find its scalar potential. (Predicted)
Q.5) Evaluate \( \int_C \vec{F} \cdot d\vec{r} \) using Green's theorem. (Predicted)
