How would you solve this differential equation?
dy/dx=1/2x+y-1
It is not clear whether the equation is
dy/dx = 1/(2x+y-1)
or
dy/dx = x/2 + y -1 (as you have written)
Please clarify.
x/2 + y - 1
To solve the differential equation dy/dx = 1/2x + y - 1, we can use the method of integrating factors. Here's how you can do it step by step:
Step 1: Rewrite the differential equation in the standard form.
dy/dx + (-1/2)x - y = -1
Step 2: Identify the coefficients of dy/dx, x, and y.
In this case, the coefficient of dy/dx is 1, the coefficient of x is -1/2, and the coefficient of y is -1.
Step 3: Multiply the entire equation by an integrating factor.
The integrating factor is defined as exp∫P(x)dx, where P(x) is the coefficient of x. In this case, P(x) = -1/2.
So, the integrating factor is exp(∫(-1/2)dx) = exp(-1/2x) = e^(-1/2x).
Multiplying both sides of the equation by e^(-1/2x) gives us:
e^(-1/2x) * dy/dx + (-1/2)x * e^(-1/2x) * y = -e^(-1/2x)
Step 4: Simplify the left side of the equation.
Using the product rule for differentiation, the left side can be written as:
d/dx (e^(-1/2x) * y) = -e^(-1/2x)
Step 5: Integrate both sides of the equation.
Integrating both sides with respect to x yields:
∫d/dx (e^(-1/2x) * y) dx = ∫(-e^(-1/2x)) dx
Integrating the left side gives us:
e^(-1/2x) * y = 2e^(-1/2x) + C
where C is the constant of integration.
Step 6: Solve for y.
To solve for y, we divide both sides of the equation by e^(-1/2x), giving:
y = 2 + Ce^(1/2x)
So, the solution to the given differential equation is y = 2 + Ce^(1/2x), where C is any constant.