Differential Equations Answers

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

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

Apply the method of variations of parameters to solve the following differential equations:
1) x^2 y" + xy' - y = x^2 e^x
2) y" + a^2 y = cosec ax
3) solve the equation d^2 y/dx^2 - cotx dy/dx - sin^2 xy = cosx - cos^3 x by changing the independent variable

solve the following equation by changing the independent variable x^2 d^2 y/dx - dy/dx -4x^3 y = 8x^3 sin x^2 , x>0

1.1) y^-2 dy/dx + y^-1 = 2x

1.2) (y^2e^xy^2 +4x^3)dx + (2xye^xy^2 - 3y^2)dy = 0

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 C
o
290 and the substance cools from
C
o
370 to C
o
330 in 10 minutes, find when the temperature will be C
o
295

Solve the ordinary differential equation
d2y/dx2+dy/dx+y=0

Let y1 and y2 be two solutions of the equation a2(x)y'' + a1(x)y' + a0(x)y =0. If W(y1, y2) is the Wronskian of y1 and y2, show that. a2(x)(dw/dx) + a1(x)W = 0

A mass weighing 39.5 kg. stretches a spring 1/4m. At t=0 , the mass is released from a point 3/4m below the equilibrium position with an upward velocity of (5/4)m/second. Determine the function x(t) that describes the subsequent free motion.

Solve the differential equation:
xdy - (3y + x^5. y^1/3)dx = 0