Differential Equations Answers

u_tt =c²u_xx=, 0<x<1

u(0,1)=0, t >0

u(1, t)=0, t >0

u(x, 0)= 2x(1-x), (0<=x <= 1)

u_t (x, 0) =0

For high-speed motion through the air—such as the skydiver shown in the figure below, falling before the parachute is opened—air resistance is closer to a power of the instantaneous velocity v(t).

Determine a differential equation for the velocity v(t)

 of a falling body of mass m if air resistance is proportional to the square of the instantaneous velocity. Assume the downward direction is positive. (Use k > 0

 for the constant of proportionality, g > 0

 for acceleration due to gravity, and v for v(t).)

Solve the initial value probelm of y'-y=e^x, y(1)=0

Find the initial value problem of y'=y+x/y-x, y(0)= 2

In the following one solution of a second y1 order linear homogene DE is given. Find the second linearly independent solution y2 using the method of reduction of order.

2x²y" + 3xy'-y=0 ,. y1=(1/x)

(1-x²)y"-2xy'+2y=0 , y1=x

x²y"+2xy'-2y=0,. y1=x

x²y"+3xy'+y=0, y1=1/x

x²y"-x(x+2)y'=0,. y1=x

In the following one solution of a second y1 order linear homogene DE is given. Find the second linearly independent solution y2 using the method of reduction of order.

1. 2x^²y" + 3xy'-y=0 ,.
2. (1-x²)y"-2xy'+2y=0 ,
3. x²y"+2xy'-2y=0,
4. x²y"+3xy'+y=0,
5. x²y"-x(x+2)y'=0,

xy"+2y'=0 reduced into y

y"=2y' reducing this into y

xdy + (y - x² y² ) dx = 0

xdy + (y-x^2y^2)dx = 0