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Differential Equations
Given that z = ax + √a² − 4y + c
2
is the complete integral of the PDE,
p² − q² =4
determine its general integral.
2
is the complete integral of the PDE,
p² − q² =4
determine its general integral.
Differential Equations
According to Newton’s law of cooling, the rate at which a substance cools in moving air is proportional to the difference between the temperature of the substance and that of the air. If the temperature of the air is 290°C and the substance cools from 370°C to 330°C in 10 minutes, find when the temperature will be 295°C.
Differential Equations
iii) The p.d.e.
auxx + 2b uxy + c uyy = 0
where a, b ,c are constants is irreducible when
b 2− ac =0 .
auxx + 2b uxy + c uyy = 0
where a, b ,c are constants is irreducible when
b 2− ac =0 .
Differential Equations
The respiratory flow of air in the lungs is affected due to air pollution. If you have to model
respiratory flow write four essentials for the model.
respiratory flow write four essentials for the model.
Differential Equations
The rate of increase of susceptible AIDS victims is proportional to the number of susceptible
persons and number of infected persons. If there are 0
S susceptible persons and 1 infected
person at a time to then i) set up the equation for the spread of the disease ii) solve the resulting
equation iii) give a physical interpretation to the same by plotting the epidemic curve iv) write the
limitations of the model.
persons and number of infected persons. If there are 0
S susceptible persons and 1 infected
person at a time to then i) set up the equation for the spread of the disease ii) solve the resulting
equation iii) give a physical interpretation to the same by plotting the epidemic curve iv) write the
limitations of the model.
Differential Equations
how how to solve, by Jacobi's method, a partial differential equation of
the type f( x, delu/delx, delu/delz ) = g(y, delu/dely, delu/delz )
the type f( x, delu/delx, delu/delz ) = g(y, delu/dely, delu/delz )
Differential Equations
(a)Calculate all the four second-order partial derivatives of the following functions:
(i) f(x,y)=cos(x^2+y^2)
(ii) f(x,y)=sin(x/y)
(b) Find the range of the function f defined by f(x,y) 10–x^2–y^2 for all (x,y)
for which x^2+y^2 ≤9 Sketch two of its level curves.
(c)Check whether the following functions are differentiable at the point given
against them:
(i)f(x,y)=|x=1|at (1,0)
(ii) f(x,y) =y^3+ ysin2x +e^(x+y) at (1,–1)
(d)Find dw/dt
w = xy + x,z = cos t, y = sint , z=1 at t= 0 .
(i) f(x,y)=cos(x^2+y^2)
(ii) f(x,y)=sin(x/y)
(b) Find the range of the function f defined by f(x,y) 10–x^2–y^2 for all (x,y)
for which x^2+y^2 ≤9 Sketch two of its level curves.
(c)Check whether the following functions are differentiable at the point given
against them:
(i)f(x,y)=|x=1|at (1,0)
(ii) f(x,y) =y^3+ ysin2x +e^(x+y) at (1,–1)
(d)Find dw/dt
w = xy + x,z = cos t, y = sint , z=1 at t= 0 .
Differential Equations
(i)Find the integral of f(x,y ) x^4+y^2 over the region bounded by y=x , y= 2x and x=2
(ii)Find the surface area of the portion of the paraboloid z=25–x^2–y^2 which lies
above the xy -plane.
(iii)Locate and classify the stationary points of the function
f( x,y ) x^2+y^2–6xy +6x +3y –4
(iv)Check whether the following functions are homogeneous or not.
(a)x/y +3y/2x +sin√(x/y)
(b)x^4 +4x^2 +y^2)x^2
(v)Evaluate f(xy )at a point (x,y)for the function f defined by
f(x,y)=x^5 +10x^3 y^3 +8y^4
Verify that the function f satisfies the requirements of Schwarz’s theorem and
hence evaluate f(x,y)
(ii)Find the surface area of the portion of the paraboloid z=25–x^2–y^2 which lies
above the xy -plane.
(iii)Locate and classify the stationary points of the function
f( x,y ) x^2+y^2–6xy +6x +3y –4
(iv)Check whether the following functions are homogeneous or not.
(a)x/y +3y/2x +sin√(x/y)
(b)x^4 +4x^2 +y^2)x^2
(v)Evaluate f(xy )at a point (x,y)for the function f defined by
f(x,y)=x^5 +10x^3 y^3 +8y^4
Verify that the function f satisfies the requirements of Schwarz’s theorem and
hence evaluate f(x,y)
Differential Equations
Obtain the differential equation associated with y= Be^x with A&B being arbitrary constants
Differential Equations
Classify the following statements as true or false giving reasons for your answers. i) General solution of the differential equations x^2(d^2y/dx^2)^(3) +y(dy/dx)^(4) +y^4 =0 must contain four arbitrary constants.
(ii) The set of real (or complex) solutions of equation y'(x) + P(x)y(x)=Q(x) forms a
real (or complex) vector space.
(iii) sin xd^2y/dx^2 +dy/dx +y=0 ]0,π[ is linear homogeneous equation.
iv) The partial differential equation
x^2 ∂^2z/∂x^2 +2xy ∂^2z/∂x∂y +y^2 ∂^2z/∂y^2 –x^m y^n =0
is a reducible homogeneous equation.
(v) The partial differential equation u ∂u/∂x =e^y + sin x, u= u(x,y) is a quasi-linear
p.d.e.
(ii) The set of real (or complex) solutions of equation y'(x) + P(x)y(x)=Q(x) forms a
real (or complex) vector space.
(iii) sin xd^2y/dx^2 +dy/dx +y=0 ]0,π[ is linear homogeneous equation.
iv) The partial differential equation
x^2 ∂^2z/∂x^2 +2xy ∂^2z/∂x∂y +y^2 ∂^2z/∂y^2 –x^m y^n =0
is a reducible homogeneous equation.
(v) The partial differential equation u ∂u/∂x =e^y + sin x, u= u(x,y) is a quasi-linear
p.d.e.
