7. Solve the following differential equations :3e

x
tan y dx + (1 – e
x
) sec2 y dy = 0 given that , 4
y
�
 when
x = 1.

wow. That copy/paste doesn't work too well, eh? If I can figure it out, we have

3e^x tany dx + (1+e^x)sec^2y dy = 0
(1+e^x)sec^2y dy = -3e^x tany dx

sec^2y/tany dy = -3e^x/(1+e^x) dx
1/(siny cosy) dy = -3e^x/(1+e^x) dx
2csc(2y) dy = -3e^x/(1+e^x) dx
ln(tany) = -3ln(1+e^x) + c
tany = c/(1+e^x)^3
y = arccot(c(1+e^x)^3)

Now just plug in your point to find c.

Of course, if I misinterpreted your text, you may have to modify the solution shown.

To solve the given differential equation

3e^x tan(y) dx + (1 - e^x) sec^2(y) dy = 0

we can use the method of separation of variables.

Step 1: Rearrange the equation to separate the variables dx and dy.

3e^x tan(y) dx = - (1 - e^x) sec^2(y) dy

Step 2: Divide both sides by (1 - e^x) tan(y) to get the variables on separate sides.

(3e^x / (1 - e^x)) dx = - sec^2(y) / tan(y) dy

Step 3: Integrate both sides of the equation with respect to their respective variables.

∫ (3e^x / (1 - e^x)) dx = ∫ - sec^2(y) / tan(y) dy

To find the integral of (3e^x / (1 - e^x)), we can let u = 1 - e^x:

∫ (3 / u) du = 3ln|u| + C = 3ln|1 - e^x| + C1

To find the integral of -sec^2(y) / tan(y), we can use a trigonometric identity:

-∫ sec^2(y) / tan(y) dy = -∫ (1 / cos^2(y)) / (sin(y) / cos(y)) dy
= -∫ (1 / sin(y)) dy
= -ln|sin(y)| + C2

Step 4: Set up the equation with the integrated functions.

3ln|1 - e^x| + C1 = -ln|sin(y)| + C2

Step 5: Use the given initial condition (x = 1, y = 4) to find the values of the arbitrary constants C1 and C2.

When x = 1, y = 4:
3ln|1 - e^1| + C1 = -ln|sin(4)| + C2

Since ln|1 - e^1| = ln|1 - e|, let's simplify the equation:
3ln|1 - e| + C1 = -ln|sin(4)| + C2

Step 6: Solve for C1 - C2 by moving the terms around:

3ln|1 - e| + ln|sin(4)| = C2 - C1

Step 7: Simplify the equation further:

ln|sin(4)| - ln|1 - e| = C2 - C1

Step 8: Combine the logarithmic terms:

ln|(sin(4)) / (1 - e)| = C2 - C1

So, the general solution to the given differential equation is:

3ln|1 - e^x| + ln|sin(y)| = ln|(sin(4)) / (1 - e)| + C

where C = C2 - C1 represents the constant of integration.