Differential equation 1 marks

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

(A) = e – 4+ 5

(B) = e – 4+ 1.25

(C) y= e 4+ 5

(D) y= e 4+ 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) \(\large (1-x)e^{-x^2}\)


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