Differential Equations Answers

Solve the differential equation : (xy^2 - x^2) dx +(3x^2y^2 +x^2y-2x^3+y^2) dy =0.

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

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

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)

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

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.

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

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.

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

Find the value of b for which the equation

(ye^2xy + x) dx + (bxe^2xy) dy =0

is exact, and hence solve it for that value of b


b) Find the solution of the Riccati equation

dy/dx =2cos^2x-sin^2x+y^2 / 2cosx

y1(x) = sin x