find the solution to the following differential equations I

find the solution to the following differential equations

I need all steps

find the solution to the following differential equations (a)dH/dz(z) + piH(z) = 0, with H(0) = 1. Answer: H(1) = e^-pi (b) dy/dt(t) - sigmay(t) = 0, with y(1) = pi Answer: with sigma = 1,y(2) = 8.54. (c) dx/dy(y) = ax(y), withx(0) = 2. Answer: with a = 2, x(1) = 14.8. I need all steps

Solution

dH/dz = -pi * H

dH/H = -pi * dz

Integrating :

ln|h| = -pi*z + C

H(0) = 1 :
ln|1| = -pi * 0 + C
C = 0

So, ln|h| = -pi * z

h = e^(-pi * z)

h(1) = e^(-pi * 1)

h(1) = e^(-pi) ----> ANSWER

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I shal refer to \"sigma\" as \'s\'

dy/dt - sy = 0
dy/dt = sy
dy/y = s*dt

Integrating :

ln|y| = st + C

y(1) = pi :

ln|pi| = s + C

Given s = 1 :

ln(pi) = 1 + C
C = ln(pi) - 1

So,
ln|y| = st + C
becomes
ln|y| = 1t + ln(pi) - 1
y = e^(t - 1) * e^ln(pi))
y = pi * e^(t - 1)

So,
y(2) = pi * e^(2 - 1)
y(2) = pi * e
y(2) = 8.54 approx ---> SECOND ANSWER

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dx/dy = ax
dx/x = a*dy
Integrating :

ln(x) = ay + C

x = e^(ay + C)

x = De^(ay)

x(0) = 2 :

2 = De^(0)
D = 2

x = 2e^(ay)

When a = 2,
x = 2e^(2y)

x(1) = 2e^(2*1)
x(1) = 2e^2
x(1) = 14.78 ---> THIRD ANSWER