Find m and b so that y=mx+b is a solution to the differential equation
dy/dx=1/2x+y-1
dy/dx=(1/2)x + y - 1
I assume that is what you mean.
y = m x + b
dy/dx = m
so
m = .5 x + mx + b - 1
m = (.5 + m) x + b-1
well if m is to be constant, m must be -.5
that means
-.5 = b-1
b = .5
so
y = -.5 x + .5
--------the end -----
Now check with m = -.5 and b = +.5
y = -.5 x + .5
dy/dx = -.5
dy/dx = .5 x + (-.5 x +.5) - 1
= +.5 - 1
= -.5
check
I will assume you mean (1/2)x and not 1/(2x)in the differential equation
dy/dx = m, so we require that
m = x/2 + mx+ b -1
This is true if m = -1/2 and
b = 1+m = 1/2
y = -x/2 + 1/2 is a solution, but not the only solution, to the differential equation.
Well, let's put on our detective hats and solve this mystery!
First, let's rearrange the differential equation to make it easier to work with:
dy/dx - y = 1/2x - 1
Now, we can see that this is a first-order linear ordinary differential equation of the form:
dy/dx + P(x)y = Q(x)
where P(x) = -1 and Q(x) = 1/2x - 1.
To solve this type of differential equation, we can use an integrating factor, which is given by the formula:
μ(x) = e^(∫P(x)dx)
In this case, our integrating factor, μ(x), is given by:
μ(x) = e^(∫-1dx) = e^(-x)
Now, multiplying both sides of our rearranged equation by μ(x), we get:
e^(-x) * dy/dx - e^(-x) * y = e^(-x) * (1/2x - 1)
Using the product rule on the left-hand side, we have:
d/dx [e^(-x) * y] = e^(-x) * (1/2x - 1)
Integrating both sides with respect to x, we get:
∫d/dx [e^(-x) * y] dx = ∫e^(-x) * (1/2x - 1) dx
Simplifying the left-hand side using the fundamental theorem of calculus, we have:
e^(-x) * y = ∫e^(-x) * (1/2x - 1) dx
To integrate the right-hand side, we can break it up into two parts:
∫e^(-x) * (1/2x) dx - ∫e^(-x) dx
The first integral can be evaluated using integration by parts, and the second integral can be evaluated using the fact that ∫e^(-x) dx = -e^(-x).
After solving for y in terms of x, we obtain the general solution to the differential equation:
y = (1/2 + C) * e^x - 1
where C is the constant of integration.
Now, since we want y = mx + b to be a solution to the given differential equation, we can equate the two equations:
(1/2 + C) * e^x - 1 = mx + b
To find m and b, we need to match coefficients:
m = 1/2 + C
b = -1
And there you have it! The values for m and b are m = 1/2 + C and b = -1.
Remember, comedy may not be my strong suit, but solving differential equations is a mystery I can unravel with ease!
To find m and b so that y = mx + b is a solution to the given differential equation dy/dx = (1/2)x + y - 1, we need to substitute y = mx + b into the differential equation and solve for m and b.
Let's start by differentiating y = mx + b with respect to x to find dy/dx:
dy/dx = d/dx (mx + b)
= m
Now, replace y and dy/dx in the given differential equation with mx + b and m, respectively:
m = (1/2)x + (mx + b) - 1
Next, simplify the equation:
m = (1/2)x + mx + b - 1
Combine like terms:
m = (1/2)x + mx - 1 + b
Rearrange the equation to isolate terms with m and b together:
m - mx = (1/2)x + b - 1
Factor out m and b:
m(1 - x) = (1/2)x + (b - 1)
Now we have two equations:
1) m(1 - x) = (1/2)x + (b - 1)
2) m = (1/2)x + y - 1
We can solve this system of equations to find the values of m and b.
First, let's solve equation 2) for y:
y - 1 = m - (1/2)x
y = m - (1/2)x + 1
Substitute this expression for y into equation 1):
m(1 - x) = (1/2)x + (b - 1)
m - mx = (1/2)x + b - 1
Now we have two equations with m and b:
1) m - mx = (1/2)x + b - 1
2) m = m - (1/2)x + 1
We can simplify these equations:
m - mx = (1/2)x + b - 1 [Equation 1]
m + (1/2)x - 1 = m [Equation 2]
Subtract m from both sides of equation 2):
(1/2)x - 1 = 0
Now add mx to both sides of equation 1):
m = (1/2)x + b - 1 + mx
m = (1/2)x + b - 1 + (1/2)mx
Combine like terms:
m - (1/2)mx = (3/2)x + b - 1
Factor out m:
m(1 - (1/2)x) = (3/2)x + b - 1
Set the coefficients of m on both sides equal to each other:
1 - (1/2)x = 0
Solve for x:
(1/2)x = 1
x = 2
Now substitute x = 2 into equation 2):
(1/2)(2) - 1 = 0
1 - 1 = 0
0 = 0
Since this equation is always true, we have an identity, which means that for any value of x, we can choose any values for m and b, and the equation will hold true.
Therefore, we can choose any values for m and b that we want, and y = mx + b will be a solution to the given differential equation dy/dx = (1/2)x + y - 1.
To find the values of m and b such that y = mx + b is a solution to the differential equation dy/dx = (1/2)x + y - 1, we need to substitute the given solution into the differential equation and see if it satisfies the equation.
Let's start by finding dy/dx of y = mx + b. Taking the derivative of y with respect to x, we get:
dy/dx = d/dx (mx + b)
= m
Now, substitute y = mx + b and dy/dx = m into the differential equation:
m = (1/2)x + mx + b - 1
Simplifying the equation:
m - mx = (1/2)x - b + 1
Factoring out the common terms on both sides:
m(1 - x) = (1/2)x - (b - 1)
To satisfy the equation for all values of x, both sides of the equation must be equal. Thus, we have:
m = (1/2)
1 - x = -(b - 1)
From the first equation, we can determine that m = 1/2.
Substituting m = 1/2 into the second equation:
1 - x = -(b - 1)
Simplifying the equation:
1 - x = -b + 1
Move the terms to the respective sides:
x - 1 = b - 1
Cancel out the -1 terms:
x = b
Therefore, the values of m and b that satisfy the differential equation dy/dx = (1/2)x + y - 1 for the solution y = mx + b are m = 1/2 and b = x.