Answer:
To find a solution, we are given that f(x) = 1 is a solution. This means that when x = f(x) = 1, the equation is satisfied.
Let's substitute x = 1 into the differential equation:
dy/dx = (1 - 1)y^2 + (2(1) - 1)y - 1
dy/dx = 0 + 1y - 1
dy/dx = y - 1
Since the left side and right side are equal, we have:
y - 1 = y - 1
This confirms that f(x) = 1 is indeed a solution to the given differential equation.
Now, let's solve the differential equation step by step.
Separating variables, we can rewrite the equation as:
dy/(y - 1) = dx
Next, let's integrate both sides:
∫dy/(y - 1) = ∫dx
This gives us:
ln|y - 1| = x + C
where C is the constant of integration.
Now, let's solve for y:
Taking the exponential of both sides, we have:
e^(ln|y - 1|) = e^(x + C)
Simplifying, we get:
|y - 1| = e^(x + C)
Since e^(x + C) is always positive, we can remove the absolute value sign:
y - 1 = e^(x + C)
To simplify further, let's rewrite e^C as a constant:
y - 1 = Ce^x
Finally, adding 1 to both sides, we obtain the general solution:
y = Ce^x + 1
where C is an arbitrary constant.
In conclusion, the solution to the given differential equation dy/dx = (1 - x)y^2 + (2x - 1)y - x, with the initial condition f(x) = 1, is y = Ce^x + 1, where C is an arbitrary constant.
Have a great day/week.
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