(2y-1)dy/dx=(3x^2+1) given that x=1 when y=2
plz show step thanks
huh? simple separation of variables.
(2y-1) dy = (3x^2+1) dx
y^2-y = x^3+x + C
Using your point (1,2), find C:
4-2 = 1+1+C
C = 0
y^2-y = x^3+x
y = (1±√(4x^3+4x+1))/2
To solve the given differential equation (2y - 1)dy/dx = (3x^2 + 1), follow these steps:
Step 1: Separate the variables.
Rearrange the equation to separate the y terms on one side and the x terms on the other side.
(2y - 1) dy = (3x^2 + 1) dx
Step 2: Integrate both sides.
Integrate both sides of the equation by integrating the left side with respect to y and the right side with respect to x.
∫(2y - 1) dy = ∫(3x^2 + 1) dx
Step 3: Perform the integrations.
On the left side, integration of (2y - 1) with respect to y gives y^2 - y.
On the right side, integration of (3x^2 + 1) with respect to x gives x^3 + x.
Step 4: Add a constant of integration.
Since we have integrated both sides, we need to add a constant of integration (C) on the right side as the result of the integration.
y^2 - y = x^3 + x + C
Step 5: Use the initial condition.
Use the given initial condition x = 1 when y = 2 to determine the value of the constant of integration. Plug in x = 1 and y = 2 into the equation.
(2^2 - 2) = 1^3 + 1 + C
4 - 2 = 2 + 1 + C
2 = 3 + C
C = -1
Step 6: Substitute the value of C back into the equation.
Substitute C = -1 into the equation.
y^2 - y = x^3 + x - 1
So, the solution to the given differential equation is y^2 - y = x^3 + x - 1.
To solve the given differential equation, we can use the method of separation of variables. Here are the step-by-step calculations:
Step 1: Separate the variables by moving all y terms to one side and all x terms to the other side. The given equation can be rewritten as:
(2y - 1) dy = (3x^2 + 1) dx
Step 2: Integrate both sides with respect to their respective variables. Integrating the left side with respect to y and the right side with respect to x, we get:
∫ (2y - 1) dy = ∫ (3x^2 + 1) dx
Step 3: Perform the integrals on each side separately. The integral of (2y - 1) with respect to y is:
∫ (2y - 1) dy = y^2 - y + C1 (where C1 is the constant of integration)
The integral of (3x^2 + 1) with respect to x is:
∫ (3x^2 + 1) dx = x^3 + x + C2 (where C2 is another constant of integration)
Step 4: Set the integrated expressions equal to each other:
y^2 - y + C1 = x^3 + x + C2
Step 5: Apply the initial condition (x = 1, y = 2) to find the specific values of the constants C1 and C2. Substituting these values into the equation, we get:
2^2 - 2 + C1 = 1^3 + 1 + C2
4 - 2 + C1 = 2 + C2
C1 + 2 = C2 (equation 1)
Step 6: Rewriting the equation with the constant C2 eliminated, we have:
y^2 - y + C1 = x^3 + x + (C1 + 2)
y^2 - y = x^3 + x + 2
Step 7: Simplify and rearrange the equation to the desired form:
y^2 - y - x^3 - x - 2 = 0
Now you have the final form of the equation, which represents the solution to the given differential equation.