Which of the following is a separable, first-order differential equation?
A) dy/dx= x+y/2x
B) dy/dx=x+y/x-y
C) dy/dx=sinx <<
D) dy/dx=xy/x+y
E) dy/dx= 2lnxy
2) The differential equation dy/dx=y/x^2 has a solution given by:
A) y=C sqrt(x^2+4)
B)y=C e^-1/x
C)-1/2ex^2+C
D) x^2+y^2=C
E)y=Ce^-kt
1C
A) "dy/dx= x+y/2x" That equation looks like a complicated relationship status. It's not separable, but it might need some self-reflection and personal growth.
B) "dy/dx=x+y/x-y" This equation seems like a problematic relationship. It's not separable, just like the couple involved.
C) "dy/dx=sinx" Ah, this one seems like the odd one out. It's definitely separable, just like the tendency of people to separate themselves from the dance floor when the song "sin(x)" comes on.
D) "dy/dx=xy/x+y" This equation seems a bit clingy, not separable at all. It's like two people who are just too dependent on each other.
E) "dy/dx= 2lnxy" This equation is logarithmic, but it's not separable. It's like trying to separate the ingredients in a complicated recipe. It's just too mixed up.
So, the correct answer to the first question is C) dy/dx=sinx.
As for the second question,
A) "y=C sqrt(x^2+4)" This equation suggests that love is a mathematical equation, and that C is the constant amount of romantic feelings you have for someone.
B) "y=C e^-1/x" This equation seems like a romantic relationship that diminishes rapidly with any amount of distance. It's as if love turns into a complex exponential function in this case.
C) "-1/2ex^2+C" This equation represents a curve that goes up and down, just like the roller coaster of emotions in a relationship. It seems like a possible solution.
D) "x^2+y^2=C" This equation represents a circle, which doesn't really match the given differential equation. It's like trying to fit a square peg into a circular relationship.
E) "y=Ce^-kt" This equation suggests a relationship that fades away exponentially with time, like a love that slowly disappears. It's possible, but not the correct solution.
Therefore, the correct answer to the second question is C) -1/2ex^2+C.
Remember, it's all equations and humor here. Don't take it too seriously!
1) The separable, first-order differential equation is option C) dy/dx = sinx.
To solve this equation, we can separate the variables by bringing all terms involving y on one side and all terms involving x on the other side.
Starting with the given equation:
dy/dx = sinx
Rearranging the equation:
dy = sinx dx
Integrating both sides with respect to their respective variables:
∫ dy = ∫ sinx dx
The integral of dy with respect to y is simply y, and the integral of sinx with respect to x is -cosx (plus a constant of integration C).
Therefore, the solution to the separable, first-order differential equation is:
y = -cosx + C, where C is the constant of integration.
2) The solution to the differential equation dy/dx = y/x^2 is option B) y = Ce^(-1/x).
To solve this equation, we can separate the variables and integrate both sides.
Starting with the given equation:
dy/dx = y/x^2
Separating variables by bringing all terms involving y on one side and all terms involving x on the other side:
dy/y = dx/x^2
Integrating both sides with respect to their respective variables:
∫ dy/y = ∫ dx/x^2
The integral of dy/y is ln|y| (plus a constant of integration C1), and the integral of dx/x^2 is -1/x (plus a constant of integration C2).
Therefore, the solution to the differential equation is:
ln|y| = -1/x + C, where C is the constant of integration.
Exponentiating both sides to eliminate the natural logarithm:
|y| = e^(-1/x + C)
Since the absolute value of y is involved, we can remove the absolute value by considering two cases:
Case 1: y > 0, then |y| = y.
y = e^(-1/x + C)
y = e^C * e^(-1/x)
y = Ce^(-1/x), where C = e^C.
Case 2: y < 0, then |y| = -y.
-y = e^(-1/x + C)
y = -Ce^(-1/x), where C = -e^C.
Combining both cases, we get the general solution:
y = Ce^(-1/x), where C is any constant.
To determine which of the given options is a separable, first-order differential equation, we need to understand the characteristics of such equations.
A separable differential equation is one that can be written in the form of:
dy/dx = f(x) * g(y),
where f(x) is a function of x only and g(y) is a function of y only.
A first-order differential equation is one that involves only the first derivative of the dependent variable y with respect to the independent variable x.
Let's check each option to see which one meets these criteria.
Option A) dy/dx = x + y/2x:
This equation is not separable since both the function f(x) and g(y) are involved in the equation.
Option B) dy/dx = x + y/x - y:
Again, this equation is not separable as both f(x) and g(y) appear together.
Option C) dy/dx = sin(x):
This option is a separable, first-order differential equation:
dy/dx = sin(x) can be rewritten as dy = sin(x) dx.
Option D) dy/dx = xy/x + y:
This equation is not separable as both f(x) and g(y) are combined.
Option E) dy/dx = 2ln(xy):
This equation is not a first-order differential equation since it involves both the first derivative and the product of x and y.
Therefore, the answer is (C) dy/dx = sin(x), as it is the only separable, first-order differential equation among the given options.
Moving on to the second question:
The differential equation dy/dx = y/x^2 can be solved by separating the variables and integrating both sides.
Dividing both sides by y, we have:
(1/y) dy/dx = 1/x^2.
Now, we can rewrite this as:
dy/y = dx/x^2.
Integrating both sides separately, we have:
∫(1/y) dy = ∫(1/x^2) dx.
The left integral becomes ln|y| + C1, where C1 is the constant of integration.
The right integral becomes -1/x + C2, where C2 is the constant of integration.
Therefore, our equation becomes:
ln|y| + C1 = -1/x + C2.
Rearranging and combining constants, we get:
ln|y| = -1/x + K,
where K = C2 - C1 is a new constant.
Finally, exponentiating both sides gives us:
|y| = e^(-1/x + K).
Taking into account the absolute value, we can write the solution as:
y = ± e^(K) * e^(-1/x).
Since e^(K) is a positive constant, we can replace it with another constant, C, to simplify:
y = C * e^(-1/x),
where C is the constant of integration.
Therefore, the solution to the differential equation dy/dx = y/x^2 is given by option (B) y = C e^(-1/x).
#1 ok
#2
dy/y = dx/x^2
lny = -1/x + c
y = e^(-1/x + c)
= ce^(-1/x)
so, (B)