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The volume of an object expressed in spherical coordinates is given by
The value of the integral is
By a change of variable x (u, y) = uv, y (u, v) = v/u is double integral, the integrand f(x, y) changes to f(uv, v/u) φ (u,v). Then, φ (u, v) is
Consider the shaded triangular region P shown in the figure. What is
The equation of the line AB is
A path AB in the form of one quarter of a circle of unit radius is shown in the figure. Integration of (x + y)^{2} on path AB traversed in a counterclockwise sense is
A parabolic cable is held between two supports at the same level. The horizontal span between the supports is L. The sag at the midspan is h. The equation of the parabola is where x is the horizontal coordinate and y is the vertical coordinate with the origin at the centre of the cable. The expression for the total length of the cable is
We know length of the curve f(x) between x = a and x = b given by
A surface S(x,y)=2x+5y3 is integrated once over a path consisting of the points that satisfy ( x +1)2+ (y − 1)2 = √2 . The integral evaluates to
The order of the differential equation
The order of a differential equation is the order of the highest derivative involving in
equation, so answer is 2.
The solution of the differential equation under the boundary conditions
(i) y =y_{1} At x = 0 and
(ii) y =y_{2} At x = ∞,
Where k, y_{1} and y_{2} are constants, is
A function n(x) satisfies the differential equation where L is a constant. The boundary conditions are: n(0)=K and n(∞)= 0. The solution to this equation is
For finite solution c_{1} = 0
The solution of the differential equation
Given differential equation is
Integra ling we get
It is linear differential equation.
The solution of dy/dx = y^{2} with initial value y (0) = 1 bounded in the interval
The solution to the differential equation f’’(x)+4f’(x)+4f(x)=0 is
Let y(x) = emx (m ≠ 0)be the trial sol^{n} .Auxiliary equation. m^{2} + 4m+ 4 = 0 ⇒(m+ 2)^{2} = 0
In particular, when A =1,B =1,then f(x) = (1 + x)e^{−2x}
= e^{−2x} + xe^{−2x}
Since, the differential equation cannot be expressed in x/y or y/x form, therefore, it is an example of nonhomogeneous differential equation.
Which of the following is a solution of the differential equation
Here p = 4andq = 3.The given equation becomes
f is non linear.
Given that .. .x+ 3x= 0, and x(0)=1, x(0) = 0 what is x(1)?
Auxiliary equn of x^{11} + 3x = 0 is
m^{2} + 3 = 0
The degree of the differential equation
The solution for the differential equation with the condition that y = 1 at x = 0 is
A spherical naphthalene ball exposed to the atmosphere loses volume at a rate proportional to its instantaneous surface area due to evaporation. If the initial diameter of the ball is 2 cm and the diameter reduces to 1 cm after 3 months, the ball completely evaporates in
By the given condition
Solution of the differential equation represents a family of
Let y = e^{mx} (m ≠ 0) be the trial solution.
A body originally at 60ºC cools down to 40ºC in 15 minutes when kept in air at a temperature of 25ºC. What will be the temperature of the body at the end of 30 minutes?
The partial differential equation that can be formed from z = ax + by + ab has the form
With K as constant, the possible solution for the first order differential equation is
The boundaryvalue problem y^{n} + λy = 0, y(0) = y(λ) = 0 will have nonzero solutions if and only if the values of λ are
The solution of the differential equation with the condition that y = 1 at x = 1, is
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