GATE 2019 Set-1

**1. For the equation \(\large \frac{\mathrm{d} y}{\mathrm{d} x}+7x^{2}y=0\),if \(y\left ( 0 \right )=\frac{3}{7}\), then the value of 𝑦(1) is**

(A) (7/3)𝑒^{−7/3 }

(B) (7/3)𝑒^{−3/7}

(C) (3/7)𝑒^{−7/3 }

(D) (3/7)𝑒^{−3/7}

GATE 2019 Set-2

**1. The differential equation d y/dx + 4y = 5 is valid in the domain 0 ≤ x ≤ 1 with y (0) = 2.25 . The solution of the differential equation is**

(A) *y *= e ^{– 4x }+ 5

(B) *y *= e ^{– 4x }+ 1.25

(C) *y*= e ^{4x }+ 5

(D) *y*= e ^{4x }+ 1.25

GATE 2018 Set-2

**1. Consider a function u which depends on position x and time t. The partial differential ****equation**

**\(\large \frac{\partial u}{\partial t}=\frac{\partial^2 u}{\partial x^2}\)**

**is known as the**

(A) Wave equation

(B) Heat equation

(C) Laplace’s equation

(D) Elasticity equation

**2. If y is the solution of the differential equation \(y^{3}\frac{\mathrm{d} y}{\mathrm{d} x}+x^{3}=0, y\left ( 0 \right )=1\) , the value of \(y\left ( -1 \right )\) is**

(A) −2

(B) −1

(C) 0

(D) 1

GATE 2017 Set-1

**1. Consider the following partial differential equation for u(x,y) with the constat c > 1 :**

**\(\large \frac{\partial u}{\partial y}+c\frac{\partial u}{\partial x}=0\)**

**Solution of this equation is**

(A) u (x,y) = f (x+cy)

(B) u (x,y) = f (x-cy)

(C) u (x,y) = f (cx+y)

(D) u (x,y) = f (cx-y)

**2. The differential equation \(\large \frac{\mathrm{d} ^2y}{\mathrm{d} x^2}+16y=0\) for y(x) with the two boundary conditions \(\large \frac{\mathrm{d} y}{\mathrm{d} x}\mid _{x=0}=1\) and \(\large \frac{\mathrm{d} y}{\mathrm{d} x}\mid _{x=\frac{\pi}{2}}=-1\) has**

(A) no solution

(B)exactly two solutions

(C) exactly one solution

(D)infinitely many solutions

GATE 2014 Set-4

**1. The solution of the initial value problem** \(\large \frac{\mathrm{d} y}{\mathrm{d} x}=-2xy;y(0)=2\)** is**

(A)\(\large 1+2e^{-x^2}\)

(B)\(\large 2e^{-x^2}\)

(C)\(\large 1+e^{x^2}\)

(D)\(\large 2e^{x^2}\)

GATE 2013

**1. The partial differential equation **\(\large\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}=\frac{\partial^2 u}{\partial x^2}\)** is a**

(A) linear equation of order 2

(B) non-linear equation of order 1

(C) linear equation of order 1

(D) non-linear equation of order 2

GATE 2010

**1.The Blasius equation,**\(\large \frac{\mathrm{d}^3f }{\mathrm{d} \eta^3}+\frac{f}{2}\frac{\mathrm{d}^2f }{\mathrm{d} \eta^2}=0\),** is a**

(A) second order nonlinear ordinary differential equation

(B) third order nonlinear ordinary differential equation

(C) third order linear ordinary differential equation

(D) mixed order nonlinear ordinary differential equation

GATE 2008

**1. Given that** \(\large \ddot{x}+3x=0\), **and** \(\large x(0)=1,\dot{x}(0)=0\), **what is** \(\large x(1)\)?

(A) -0.99

(B) -0.16

(C) 0.16

(D) 0.99

GATE 2007

**1. The partial differential equation \(\large \frac{\partial^2 \varphi}{\partial x^2}+\frac{\partial^2 \varphi}{\partial y^2}+\frac{\partial \varphi}{\partial x}+\frac{\partial \varphi}{\partial y}=0\) has**

(A) degree 1 order 2

(B) degree 1 order 1

(C) degree 2 order 1

(D) degree 2 order 2

GATE 2006

**1. The solution of the differential equation \(\large \frac{\mathrm{d} y}{\mathrm{d} x}+2xy=e^{-x^2}\) with y(0)=1 is:**

(A) \(\large (1+x)e^{+x^2}\)

(B) \(\large (1+x)e^{-x^2}\)

(C) \(\large (1-x)e^{+x^2}\)

(D) \(\large (1-x)e^{-x^2}\)