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Differential Equations
10. a) Find the surface which intersects the surfaces of the system z (x+y) = c (3z+1)
orthogonally and which passes through the circle x^2 + y^2 = 1, z = 1
b) Show that the complete integral of z = px + qy - 2p - 3q represents all possible planes
through the points (2, 3, 0)
c) Find the values of n for which the equation (n-1)^2 uxx - y^2n uyy = ny^2n-1 uy is
i) parabolic ii) hyperbolic.
orthogonally and which passes through the circle x^2 + y^2 = 1, z = 1
b) Show that the complete integral of z = px + qy - 2p - 3q represents all possible planes
through the points (2, 3, 0)
c) Find the values of n for which the equation (n-1)^2 uxx - y^2n uyy = ny^2n-1 uy is
i) parabolic ii) hyperbolic.
Differential Equations
7. a) Solve the following equations:
(ii) (D^2 - 2DD' + D'^2) z = tan (y+x)
b) Solve: z (p-q) = z^2 + (x+y^2)
(ii) (D^2 - 2DD' + D'^2) z = tan (y+x)
b) Solve: z (p-q) = z^2 + (x+y^2)
Differential Equations
6. Find the integral curves of the following equations:
a) dx / x^2 - y^2 - yz = dy / x^2 - y^2 - zx = dz / z (x - y)
b) dx / x^2 + y^2 = dy / 2xy = dz / z (x + y)
c) Find the integral surface of the partial differential equation
(x - y) y^2 p + (y - x) x^2 q = (x^2 + y^2 ) z
through the curve xz = a^2, y = 0
a) dx / x^2 - y^2 - yz = dy / x^2 - y^2 - zx = dz / z (x - y)
b) dx / x^2 + y^2 = dy / 2xy = dz / z (x + y)
c) Find the integral surface of the partial differential equation
(x - y) y^2 p + (y - x) x^2 q = (x^2 + y^2 ) z
through the curve xz = a^2, y = 0
Differential Equations
5. Apply the method of variations of parameters to solve the following differential equations:
a) x^2 y'' + x y' - y = x^2 e^x
b) y'' + a^2 y = cosec ax
c) Solve the equation d^2 y / d x^2 - cot x dy/dx - sin^2 xy = cos x - cos^3 x by changing the independent
variable.
a) x^2 y'' + x y' - y = x^2 e^x
b) y'' + a^2 y = cosec ax
c) Solve the equation d^2 y / d x^2 - cot x dy/dx - sin^2 xy = cos x - cos^3 x by changing the independent
variable.
Differential Equations
4. a) A wet porus substance in the open air loses its moisture at a rate proportional to the moisture
content. If a sheet hung in the wind loses half its moisture during the first hour, then find the
time when it has lost 95% moisture provided the weather conditions remain the same.
b) If the interest is compounded continuously, the amount of money invested increases at a rate
proportional to its size. Let Rs. 10,000 be invested at 10% compounded continuously. Then in
how many years will the original investment double itself ?
c) Solve: dy/dx + (x/1-x^2) y = xy, y(0)=1.
content. If a sheet hung in the wind loses half its moisture during the first hour, then find the
time when it has lost 95% moisture provided the weather conditions remain the same.
b) If the interest is compounded continuously, the amount of money invested increases at a rate
proportional to its size. Let Rs. 10,000 be invested at 10% compounded continuously. Then in
how many years will the original investment double itself ?
c) Solve: dy/dx + (x/1-x^2) y = xy, y(0)=1.
Differential Equations
3. d) A mass m (in kgs) acted on by a constant force p Newtons, moves a distance x in t sec.
and acquires a velocity v m/ s . Show that
x = mv^2 / 2gp = gt^2p / 2m
where g is the acceleration due to gravity.
and acquires a velocity v m/ s . Show that
x = mv^2 / 2gp = gt^2p / 2m
where g is the acceleration due to gravity.
Differential Equations
3. a) Write a suitable form of particular solution for solving the equation
y'' + 3y' + 2y = e^x (x^2 + 1) sin 2x + 3e^-x cos x + 4e^x
by the method of undetermined coefficients.
b) Verify that e^x and xe^x are solutions of the homogeneous equation corresponding to the
equation
y''− 2y'+ y = e^x / (1 + x^2), −&<x<&
and find the general solution using the method of variation of parameters.
c) If y1 = 2x+2 and y2 = -x^2 / 2 are the solutions of
xy' + y' - y'^2 / 2
then are the constant multiples c1 y1 and c2 y2 , where c1 and c2 are arbitrary, also solutions
of the given differential equation? Is the sum y1 + y2 a solution? Justify your answer.
y'' + 3y' + 2y = e^x (x^2 + 1) sin 2x + 3e^-x cos x + 4e^x
by the method of undetermined coefficients.
b) Verify that e^x and xe^x are solutions of the homogeneous equation corresponding to the
equation
y''− 2y'+ y = e^x / (1 + x^2), −&<x<&
and find the general solution using the method of variation of parameters.
c) If y1 = 2x+2 and y2 = -x^2 / 2 are the solutions of
xy' + y' - y'^2 / 2
then are the constant multiples c1 y1 and c2 y2 , where c1 and c2 are arbitrary, also solutions
of the given differential equation? Is the sum y1 + y2 a solution? Justify your answer.
Differential Equations
y=2x^3-9x^2+12x+5 it has minima at x=2 and maxima at x=1. my question is any other maxima or minima can be found except those point?
Differential Equations
y=x+(1/x). can we find any other maxima or minima at any other point except x=1 and -1.i sketch the graph , at x=1 there is a minima and x=-1 there is a maxima . but can not understand from graph why the maxima is less than the minima.
Differential Equations
what can be the graph of a curve where there is no maxima and minima?
