x³dy/dx + 3y² = xy²
Given x=1 when y=1
plz help me show step help!!
x^3 dy/dx + 3y^2 = xy^2
x^3 dy = y^2(x-3) dx
dy/y^2 = (x-3)/x^3 dx
-1/y = -1/x + 3/(2x^2) + c
1/y = (3-2x)/(2x^2) + c
y = 2x^2/(2x-3+cx^2)
y(1) = 1, so
2/(c-1) = 1
c = 3
y = 2x^2/(3x^2+2x-3)
Sure, I'd be happy to help you show the steps! Let's begin by separating the variables to make it easier to solve the differential equation:
x³(dy/dx) + 3y² = xy²
First, let's rewrite it as:
x³(dy/dx) = xy² - 3y²
Now, let's divide both sides of the equation by (xy² - 3y²):
(dy/dx) = (xy² - 3y²) / x³
Next, we can simplify by factoring out a common factor of y²:
(dy/dx) = y²(x - 3) / x³
Now, let's separate the variables to make the equation easier to solve. We'll move all terms involving y to one side and all terms involving x to the other side:
(dy / y²) = (x - 3) / x³ dx
Next, let's integrate both sides of the equation:
∫(dy / y²) = ∫((x - 3) / x³) dx
To integrate the left-hand side, we can use the power rule:
∫(dy / y²) = -1/y
For the right-hand side, we need to simplify the integrand first:
(x - 3) / x³ = (x / x³) - (3 / x³) = 1/x² - 3/x³
∫((x - 3) / x³) dx = ∫ (1/x² - 3/x³) dx
∫ (1/x² - 3/x³) dx = -1/x - 3/(2x²) + C
Now, let's combine the integrals:
-1/y = -1/x - 3/(2x²) + C
To solve for y, we can multiply both sides of the equation by -1:
1/y = 1/x + 3/(2x²) - C
Finally, let's take the reciprocal of both sides to solve for y:
y = 1 / (1/x + 3/(2x²) - C)
Now, we can substitute the initial condition x = 1, y = 1 into the equation to find the value of C:
1 = 1 / (1/1 + 3/(2*1²) - C)
1 = 1 / (1 + 3/2 - C)
1 = 1 / (5/2 - C)
To remove the fraction, we can multiply both sides by (5/2 - C):
(5/2 - C) = 1
Now, we can solve for C:
5/2 - C = 1
-C = 1 - 5/2
-C = -3/2
C = 3/2
Therefore, the solution to the differential equation is:
y = 1 / (1/x + 3/(2x²) - 3/2)
Sure! I can help you solve this differential equation step by step.
The given differential equation is: x³(dy/dx) + 3y² = xy².
To solve this, we can use the method of separation of variables. The steps involved are as follows:
Step 1: Separate the variables
We start by separating the variables. Move all terms involving y to one side and those involving x to the other side. The equation becomes:
x³(dy/dx) = xy² - 3y².
Step 2: Divide by (xy² - 3y²)
Now, divide both sides of the equation by (xy² - 3y²):
(dy/dx) = (xy² - 3y²)/x³.
Step 3: Rewrite the expression
To simplify the equation, we rewrite (xy² - 3y²) as y(x - 3y):
(dy/dx) = y(x - 3y)/x³.
Step 4: Separate the variables again
Separate the variables by multiplying both sides of the equation by dx and dividing both sides by y(x - 3y):
dy/(y(x - 3y)) = dx/x³.
Step 5: Integrate
Integrate both sides of the equation separately. The left side can be integrated using the logarithmic rule, and the right side can be integrated as a power rule. The equation becomes:
∫(dy/(y(x - 3y))) = ∫(dx/x³).
Step 6: Solve the integral
Evaluate the integrals on both sides. The left side can be evaluated as:
ln|y(x - 3y)| = -1/(2x²) + C1.
Step 7: Simplify the equation
To simplify the equation further, we remove the absolute value by taking the exponential of both sides:
y(x - 3y) = e^(-1/(2x²) + C1).
Step 8: Apply initial condition
To find the constant of integration, we can use the initial condition given in the problem, which states that when x = 1, y = 1. Substitute these values into the equation:
1(1 - 3(1)) = e^(-1/(2(1)²) + C1).
-2 = e^(-1/2 + C1).
Step 9: Solve for C1
To find the value of C1, we can take the natural logarithm of both sides:
ln|-2| = -1/2 + C1.
ln(2) = -1/2 + C1.
Step 10: Simplify C1
Solve for C1 by rearranging the equation:
C1 = ln(2) + 1/2.
Step 11: Substitute C1 back into the equation
Now that we know the value of C1, we can substitute it back into the equation we obtained in Step 7:
y(x - 3y) = e^(-1/(2x²) + ln(2) + 1/2).
Step 12: Simplify the equation
Simplify the equation further by consolidating the exponentials:
y(x - 3y) = 2e^(-1/(2x²)) * e^(1/2).
And voila! This is the final form of the solution to the differential equation.
To solve this differential equation, we can use the method of separation of variables.
Step 1: Start by rearranging the equation to isolate the terms with y and dy/dx.
x^3(dy/dx) = xy^2 - 3y^2
Step 2: Divide both sides by the expression in front of dy/dx, which is x^3.
dy/dx = (xy^2 - 3y^2)/x^3
Step 3: Now, separate the variables by multiplying both sides by dx.
dy = (xy^2 - 3y^2)/x^3 * dx
Step 4: To integrate both sides, we need to recognize that the left side is the derivative of y with respect to y, so we can integrate it as dy. On the right side, we can use algebraic manipulation to simplify the integrand.
dy = (x(y^2 - 3y^2))/x^3 * dx
dy = (y^2 - 3y^2)/x^2 * dx
dy = -2y^2/x^2 * dx
Step 5: Integrate both sides.
∫(1/y^2) dy = -2∫(1/x^2) dx
-1/y = 2/x + C
Step 6: Solve for y by isolating it.
y = -1/(2/x + C)
y = -x/(2 + Cx)
Step 7: Use the initial condition x = 1 when y = 1 to find the value of the constant C.
1 = -1/(2/1 + C)
1 = -1/(2 + C)
To solve this equation for C, multiply both sides by (2 + C).
(2 + C) = -1
C = -3
Step 8: Replace C with the value we found in step 7.
y = -x/(2 - 3x)
So, the solution to the given differential equation with the initial condition x = 1 when y = 1 is y = -x/(2 - 3x).