Differential Equations Answers

Taylors expansion of f(x) = Sin x at x = 0.

Y=e^a x.cos^3 x.sin^2 x Find dy/dx

If y=e^ax 〖cos〗^(3 ) X 〖sin〗^2 X find dy/dx

In the first order differential equation
dy/dx=f (x, y) the function f (x, y) is a function of the ratio y/x:
dy/dx=g (y/x)
Show that the substitution of U=y/x leads to separable equation in U and x.

The decay of a population by catastrophic two body collisions is described by
dN / dt =- kN2.
where 2 is supersubscribe
derive the solution.
N(t)=No (1+ t/T) -1,where o is subscribe and -1
Supersubscribe

The general solution of
d2B/dr2+dB/rdr-B/a^2 =0
is B=C Io(r a) +D Ko(r/a)
In our particular case the solution is Ko.
How we get
B= (h/2 phi a^2) Ko(r/a)
Where h= magnetic flux= Integral B d2r. Phi=3.14. And a=(m/meu. n e^2) 1/2

d/dr (dB/dr)=B/constant square.
it gave us B=r^p e^-r/constant. The power of "p" is -1/2. HOW. SHOW THAT IT IS -1\2

Solve this equation
d²Bz/dr² + dBz/rdr - Bz/a² = 0
Where a=(m/u ne²)½

dB/dx=B/t2
t is constant.if we solve this we get
B=B0 ex/t+B0e-x/t
if the graph is DECAYING EXPONENTIALY then can we ignore the B0 ex/t term?

Solve . d²Bz(x)/dx²=Bz(x)/ some contant²
The z is in subscript and solve this equation.