Solve the differential equation:
dy/dx = y^2 + y
To solve the given differential equation dy/dx = y^2 + y, we can use the method of separation of variables.
Step 1: Write the differential equation in the required form:
dy/(y^2 + y) = dx
Step 2: Separate the variables dy and dx by multiplying both sides of the equation by (y^2 + y):
dy = (y^2 + y)dx
Step 3: Integrate both sides of the equation with respect to their respective variables:
∫(1/(y^2 + y)) dy = ∫dx
Step 4: Evaluate the integrals:
To evaluate the integral on the left side, we can use a technique called partial fraction decomposition. The denominator, y^2 + y, can be factored as y(y + 1). Therefore, we can express the integrand as follows:
1/(y^2 + y) = A/y + B/(y + 1)
To find the values of A and B, we need to solve the equation:
1 = A(y + 1) + B(y)
Expanding and collecting like terms gives us:
1 = (A + B)y + A
Since the left and right sides of the equation are equal for all y, the coefficients of y must be equal on both sides:
A + B = 0
And the constant terms on both sides must also be equal:
A = 1
From the first equation, we can solve for B:
B = -A = -1
Now we can rewrite the integrand as follows:
1/(y^2 + y) = 1/y - 1/(y + 1)
Integrating both sides of the equation gives us:
∫(1/y - 1/(y + 1)) dy = ∫dx
Step 5: Evaluate the integrals:
The integral on the left side can be evaluated:
∫(1/y - 1/(y + 1)) dy = ln|y| - ln|y + 1| + C1
The integral on the right side is simply x + C2, where C2 is the constant of integration.
Therefore, our solution to the differential equation is given by:
ln|y| - ln|y + 1| = x + C
This can be further simplified using logarithmic properties:
ln(|y|/(|y+1|)) = x + C
Finally, we can simplify the absolute value expression by considering both positive and negative values:
|y|/(|y+1|) = e^(x + C)
Taking the positive and negative cases separately, we can rewrite the equation as:
y/(y+1) = Ce^x, where C is a constant of integration
This is the general solution to the given differential equation.