Test: Differential Equations & Multiple Integrals- 1
20 Questions MCQ Test GATE Mechanical (ME) 2023 Mock Test Series | Test: Differential Equations & Multiple Integrals- 1
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A triangle ABC consists of vertex points A (0,0) B(1,0) and C(0,1). The value of the integral 
over the triangle is
over the triangle is The equation of the line AB is
The area enclosed between the parabala y = x2 and the straight line y = x is
Changing the order of the integration in the double integral
When
The right circular cone of largest volume that can be enclosed by a sphere of 1 m radius has a height of
the parabolic arc 
is revolved around the x-axis. The volume of
Differential volume
What is the area common to the circles r = a and r = 2a cos θ?
Area common to circles r = a
And r = 2a cos θ is 1.228 a2
The expression 
for the volume of a cone is equal to
Choices (a) and (b) be correct because
for the volume of a cone is equal to
f ( x,y ) is a continuous defined over ( x,y ) ∈ [0,1]× [0,1] . Given two constrains, x > y 2 and y > x 2 , the volume under f ( x,y ) is
The following differential equation has
Order of highest derivative = 2
Hence, most appropriate answer is (b)
A solution of the following differential equation is given by
Let y = mx be the trial soln of the given differental equation
∴The corrosponding auxiliary equation is
For the differential equation 
the boundary conditions are
(i) y = 0 for x = 0, and
(ii) y = 0 for x = a
The form of non-zero solutions of y (where m varies over all integers) are
Here y = c1 coskx + c2 sinkx ........... (1) be the solution of the given differential equation.
Now use boundary conditions
For x = 0,y = 0 gives c1 = 0. Equation − (1) becomes
be the solution, n 0,1,2,3.......
Which of the following is a solution to the differential equation 
Hints : m + 3 = 0 ⇒m = −3
For the differential equation 
the general solution is
The solution of the differential equation 
It is a linear diff. equation
The solution of 
with the condition y(1) 
Which is 1st order linear differential equation
A technique by which problems in analysis, in particular differential equations, are transformed into algebraic problems is called
The general solution of (x2 D2 – xD), y= 0 is :
For 
the particular integrals is
It is given that y" + 2y' + y = 0, y(0) = 0, y(1)=0. What is y (0.5)?
Auxiliary equation is m2 + 2m +1 = 0⇒m = −1,−1
Using boundary condition y(0) = 0 and y(1) = 0
we get y = 0
The partial differential equation
The order is the highest numbered derivative in the equation (which is 2 here), while the degree is the highest power to which a derivative is raised (which is 1 here).
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