HPAS Maths Optional PYQs: Differential Equations

On that page, you will find year-wise and question-wise solutions.

The sources provide a comprehensive list of questions related to Differential Equations from the HPAS Maths Optional Question Paper-1 examinations. These problems cover linear and non-linear ordinary differential equations (ODEs), methods like variation of parameters, integrating factors, series solutions, and properties of special functions (Bessel and Legendre).

HPAS 2024 Differential Equations Questions

Question 1(c): Compute the general solution of the non-linear differential equation \(y = xy’ + (y’)^2\) where \(y’ = \frac{dy}{dx}\). (This is Clairaut’s form).

Question 5(a): Solve the differential equation: \(x^2\frac{d^2y}{dx^2} – 2x\frac{dy}{dx} + 2y = x + x^2\log x + x^3\). (This is a Cauchy-Euler type equation with a non-homogeneous term).

Question 5(b): Determine the power series solution about the origin of the differential equation: \((1-x^2)\frac{d^2y}{dx^2} – 4x\frac{dy}{dx} + 2y = 0\).

HPAS 2023 Differential Equations Questions

Question 5(a): Determine the general and singular solutions of the differential equation \(9p^2(2-y)^2 = 4(3-y)\), where \(p = \frac{dy}{dx}\).

Question 5(b): Solve the differential equation \(x\frac{d^2y}{dx^2} + 2\frac{dy}{dx} + \frac{xy}{2} = 0\) in terms of Bessel functions.

Question 5(c): Using the method of variation of parameters, solve the differential equation \((D^2 – 2D + 2)y = e^x \tan x\), where \(D = \frac{d}{dx}\).

Question 7(a): Let V be the vector space of real-valued functions \(y=f(x)\) satisfying \(\frac{d^3y}{dx^3} – 6\frac{d^2y}{dx^2} + 11\frac{dy}{dx} – 6y = 0\). Then show that V is a 3-dimensional vector space over \(\mathbb{R}\). (This involves solving a third-order homogeneous linear DE).

HPAS 2021 Differential Equations Questions

Question 1(d): Solve the ordinary differential equation \(xp^2 – yp – y = 0\), where \(p = dy/dx\).

Question 5(a): Show that the smallest root of the equation \(J_0(x)=0\) lies in the interval \((2, \sqrt{8})\), where \(J_0(x)\) is the Bessel’s function of order zero. (This relates to the roots of the Bessel differential equation solution).

Question 5(b): Solve the ordinary differential equation: \(\{x^2D^2 – (2m-1)xD + (m^2+n^2)\}y = n^2x^m \log x\), where \(D=d/dx\).

Question 5(c): Find the series solution near \(x=0\) of the differential equation: \(x(1-x)\frac{d^2y}{dx^2} + (1-x)\frac{dy}{dx} – y = 0\).

HPAS 2020 Differential Equations Questions

Question 1(a): Obtain the general solution of the following differential equation: \(x\frac{dy}{dx}+(2-x)y=e^{3x}\), for \(x>0\). (First Order Linear DE).

Question 7(b): Solve the following differential equation: \(x\frac{d^2y}{dx^2} – \frac{dy}{dx} = x^2e^x\), for \(x>0\).

Question 8(b): Apply the method of power series to solve the following differential equation: \((1-x^2)\frac{d^2y}{dx^2} – 2x\frac{dy}{dx} + 12y = 0\).

HPAS 2018 Differential Equations Questions

Question 1(a): Find the order and degree of the differential equation whose general solution is \(y^2 = 2c(x+\sqrt{c})\), where \(c\) is a positive parameter.

Question 2(a): Two solutions of the ordinary differential equation \(y” – 2y’ + y = 0\) are \(e^x\) and \(5e^x\). Is \(y = Ae^x + B(5e^x)\) the general solution of the differential equation?.

Question 2(b): If the integrating factor of the differential equation \((x^7y^2+3y)dx + (3x^8y-x)dy=0\) is \(x^m y^n\), then find the values of \(m\) and \(n\).

HPAS 2017 Differential Equations Questions

Question 1(c): Find the degree and order of the following differential equation: \(\left|1+\left(\frac{dy}{dx}\right)^2\right|^{2/3} = \rho\frac{d^2y}{dx^2}\).

Question 5(a): Solve the following differential equation by the method of variation of parameters: \((1-x)\frac{d^2y}{dx^2} + x\frac{dy}{dx} – y = (1-x)^2\).

Question 5(b): For Bessel’s function \(J_n(x)\), show that: \(2nJ_n(x) = x[J_{n-1}(x) + J_{n+1}(x)]\). (This is a recurrence relation derived from the Bessel DE).

HPAS 2016 Differential Equations Questions

Question 1(e): Show that the Legendre polynomial \(P_n(x)\) satisfies \(P_n(-x)=(-1)^n P_n(x)\). (This is a property related to the solution of Legendre’s DE).

Question 5(a): Determine the general and singular solution of the non-linear differential equation: \(y = xy’ + (y’)^2\).

Question 5(b): Solve the differential equation: \((D^2 – 2D + 2)y = e^x \tan x\), where \(D = \frac{d}{dx}\).

HPAS 2015 Differential Equations Questions

Question 1(d): Solve: \((D^{3}-7D-6)y = e^{2x}(1+x)\), where \(D=\frac{d}{dx}\). (Linear DE with constant coefficients).

Question 6(a): Discuss the solutions of the following equation: \(p^{2}(2-3y)^{2}=4(1-y)\).

Question 6(b): Solve: \(x^{2}\frac{d^{2}y}{dx^{2}}-(x^{2}+2x)\frac{dy}{dx}+(x+2)y=x^{3}e^{x}\).

Question 6(c): For Bessel function, prove that: \(2n~J_{n}(x)=x[J_{n-1}(x)+J_{n+1}(x)]\).

HPAS 2014 Differential Equations Questions

Question 1(e): Solve: \((D^{4}+2D^{3}-3D^{2})y=x^{3}+2~\sin~x\), where \(D\equiv\frac{d}{dx}\). (Linear DE with constant coefficients).

Question 6(a): Discuss the solutions of the following equation: \(x^{3}p^{2}+x^{2}yp+a^{3}=0\), where \(p\equiv\frac{d}{dx}\).

Question 6(b): Solve by the method of variation of parameters: \(x^{2}\frac{d^{2}y}{dx^{2}}+x\frac{dy}{dx}-y=x^{2}e^{x}\).

Question 6(c): For Bessel function, prove that: \(x~J_{n}^{\prime}(x)=n~J_{n}(x)-x~J_{n+1}(x)\).