Differential Equations Answers

1. ( y-2x)dx+xdy=0 ,y(-1)=3
2. dy=(x-4xy)dx ,y(0)= - 1/4

The integrating factor for the Bernouli differential equation 2xyy' = y^2 - 2x^3 given initial condition y(1)=2

The complementary solution for the homogeneous equation y'' + 2y' + y = 0 winll be given as

A steam pipe 20 cm in diameter contains steam at 180°C & is covered with a material 6 cm thick. If the temperature of the surface of the covering is 30°C, find the temperature halfway through the covering under steady state conditions.

A rocket is shot straight up from the earth, with a net acceleration (= acceleration by the rocket engine minus gravitational pullback) of during the initial stage of flight until the engine cut out at sec. How high will it go, air resistance neglected?

Determine a region of the xy-plane for which the given differential equation would have a unique solution whose graph passes through a point (x0, y0)in the region.

(x^2+y^2)y prime=y^2

 Consider a 1kg mass suspended to a spring. A force of 20 Newtons is required to stretch it 5 meters below it’s natural length. (a) Determine the equation describing the motion of the system, if the system is released 1 meter below it’s natural position, with a push of 5 m/s upward, and damping is such that b = 2√ 8. 

Find the integral surface of the linear differential equation (x-y)y^2p+(y-x)x^2=(x^2+y^2)z, which passes through the curve xz=a^3,y=0

Determine whether the following DE is seperable, linear , exact, homogeneous, and Bernoulli. next find its solutions by respective methods.

(1+2y^2)dy = ycosxdx

Determine whether the following Differential equation is seperable, Linear, Exact, homogeneous, and Bernoulli. Next find its solution by respective method.

8 cos^2 ydx + csc^2 xdy=0