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Differential Equations
Solve the differential equation : (xy^2 - x^2) dx +(3x^2y^2 +x^2y-2x^3+y^2) dy =0.
Differential Equations
Find a set of solutions for:
2(x-y)dx+(3x-y-1)dy=0
(2x+y-4)dx+(x-3y+12)dy=0
dy/dx=tanycotx - secycosx
2(x-y)dx+(3x-y-1)dy=0
(2x+y-4)dx+(x-3y+12)dy=0
dy/dx=tanycotx - secycosx
Differential Equations
Find a set of solutions for:
2(x-y)dx+(3x-y-1)dy=0
(2x+y-4)dx+(x-3y+12)dy=0
dy/dx=tanycotx - secycosx
2(x-y)dx+(3x-y-1)dy=0
(2x+y-4)dx+(x-3y+12)dy=0
dy/dx=tanycotx - secycosx
Differential Equations
Verify that the equations
i) z = √(2x+a) + √(2y+b) and
ii) z^2 + µ = 2(1+λ^-1) (x+λy)
are both complete integrals of the PDE .
z = 1/p + 1/q Also show that the complete integral
(ii) is the envelope of one parameter sub-system obtained by taking
b = -a/λ - µ/1+λ in the solution (i)
i) z = √(2x+a) + √(2y+b) and
ii) z^2 + µ = 2(1+λ^-1) (x+λy)
are both complete integrals of the PDE .
z = 1/p + 1/q Also show that the complete integral
(ii) is the envelope of one parameter sub-system obtained by taking
b = -a/λ - µ/1+λ in the solution (i)
Differential Equations
a) Find the differential equation of the space curve in which the two families of
surfaces
u = x^2 − y^2 = c1 and v=y^2−z^2=c2 intersect.
b) Find the general integral of the equation
( x − y)p + (y − x − z) q = z
and a particular solution through the circle z=1,x^2+y^2=1
surfaces
u = x^2 − y^2 = c1 and v=y^2−z^2=c2 intersect.
b) Find the general integral of the equation
( x − y)p + (y − x − z) q = z
and a particular solution through the circle z=1,x^2+y^2=1
Differential Equations
State whether the following statements are true or false. Justify your answer with the help of a short proof or a counter example.
i) Equation cos(x+y)p + sin(x+y)q = z^2
is a quasi-linear equation.
ii) The solution of PDE
dz/dx + dz/dy =z^2 is z = -[y+f(x-y)]
Iii)(dz/dx)(dz/dy) - (dz/dy)^2 = 0
is a non-linear PDE.
i) Equation cos(x+y)p + sin(x+y)q = z^2
is a quasi-linear equation.
ii) The solution of PDE
dz/dx + dz/dy =z^2 is z = -[y+f(x-y)]
Iii)(dz/dx)(dz/dy) - (dz/dy)^2 = 0
is a non-linear PDE.
Differential Equations
a) Identify the type of the differential equation y = xy′ +1-ln y' and hence solve it
b) Using the method of undermined coefficients, find the general solution of the differential equation
y^iv-2y'''+2y"=3e^-x + 2e^-x + e^-x sin x
c) A simple series circuit has an inductor of 1 henry, a capacitor of 10^(-6) farads and a resistor of 1000 ohms. The initial charge on the capacitor is zero. If a 12 volt battery is connected to the circuit and the circuit is closed at t =0 , find the charge on the capacitor 1 second later and the steady state charge.
b) Using the method of undermined coefficients, find the general solution of the differential equation
y^iv-2y'''+2y"=3e^-x + 2e^-x + e^-x sin x
c) A simple series circuit has an inductor of 1 henry, a capacitor of 10^(-6) farads and a resistor of 1000 ohms. The initial charge on the capacitor is zero. If a 12 volt battery is connected to the circuit and the circuit is closed at t =0 , find the charge on the capacitor 1 second later and the steady state charge.
Differential Equations
Solve, using the method of variation of parameters
d^2y/dx^2 - y = 2/1+e^x
b) Solve the following initial value problem
d^2y/dx^2 + dy/DC -2y = -6sin2x -18cos2x
y(0)=2,y'(0)=2
d^2y/dx^2 - y = 2/1+e^x
b) Solve the following initial value problem
d^2y/dx^2 + dy/DC -2y = -6sin2x -18cos2x
y(0)=2,y'(0)=2
Differential Equations
Solve the differential equationdy
dy/dx + (x/1-x^2)y = x√y, y(0)=1
b) 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.
dy/dx + (x/1-x^2)y = x√y, y(0)=1
b) 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.
Differential Equations
a) Solve the differential equation
dy/dx + xy= y^2 e^(x^2/2) sinx
b) Given that
y1(x) = x^(-1) is one solution of the differential equation
2x^2 y ′′ + 3xy′ −y =0 ,x >0,
find a second linearly independent solution of the equation
dy/dx + xy= y^2 e^(x^2/2) sinx
b) Given that
y1(x) = x^(-1) is one solution of the differential equation
2x^2 y ′′ + 3xy′ −y =0 ,x >0,
find a second linearly independent solution of the equation
