Find an equation of the curve that passes through the point (1,1) and whose slope at (x,y) is y^2/x^3.
To find an equation of the curve that passes through the point (1, 1) and has a slope at each point (x, y) given by y^2/x^3, we can use the method of separable variables.
Let's start by analyzing the slope equation: dy/dx = y^2/x^3. To solve this equation, we can separate the variables and rewrite it as:
dx = x^3 * dy / y^2
Now, we integrate both sides of the equation. On the right-hand side, the integral of dy / y^2 is -1/y, and on the left-hand side, we integrate x^3 over dx as x^4/4. Therefore, the equation becomes:
x^4/4 = -1/y + C
Here, C is the constant of integration. Since the curve passes through the point (1, 1), we can substitute these coordinates into the equation to find the value of C.
(1^4)/4 = -1/1 + C
1/4 = -1 + C
C = 1 + 1/4
C = 5/4
Substituting C = 5/4 back into the equation, we get:
x^4/4 = -1/y + 5/4
To find the equation in terms of y, we can rearrange the equation:
-1/y = x^4/4 - 5/4
Multiply both sides by -1:
1/y = 5/4 - x^4/4
Take the reciprocal of both sides:
y = 1 / (5/4 - x^4/4)
Simplify:
y = 4 / (5 - x^4)
Therefore, the equation of the curve that passes through the point (1, 1) and has a slope given by y^2/x^3 is y = 4 / (5 - x^4).