solve the differential equation dy/dx=y^2/x^3 for y=f(x) with condition y(1)=1.
dy/dx=y^2/x^3
(1/y^2) dy = (1/x^3) dx
∫(1/y^2) dy = ∫(1/x^3) dx
∫ y^-2 dy = ∫ x^-3 dx
- y^-1 = -(1/2)x^-2 + c
-1/y = -1/(2x^2) + c
at (1,1)
-1 = -1/2 + c
c = -1/2
-1/y = -1/(2x^2) - 1/2
= (-1 - x^2)/(2x^2)
1/y = (1 + x^2)/(2x^2)
y = 2x^2/(1+x^2)
f(x) = 2x^2/(1+x^2)
check my algebra, I should have written it out on paper first, did it on-screen.
That is always dangerous.
Thank you!
Hey there! Let's put on our juggling hats and solve this differential equation.
To start, we can rewrite the equation as dy/y^2 = dx/x^3. Integrating both sides, we get ∫(dy/y^2) = ∫(dx/x^3).
The integral on the left side becomes ∫(dy/y^2) = -1/y. On the right side, we have ∫(dx/x^3) = -1/(2x^2).
So now we have -1/y = -1/(2x^2) + C, where C is the constant of integration. Solving for y, we get y = -2x^2/(2x^2 - C).
To find the value of C, we can use the condition y(1) = 1. Plugging in the values, we have 1 = -2(1)^2/(2(1)^2 - C), which simplifies to 1 = -2/2 - C.
Simplifying further, we get 1 = -1 - C, and solving for C, C = -2.
So, the solution to the differential equation dy/dx = y^2/x^3 with the condition y(1) = 1 is y = -2x^2/(2x^2 + 2).
Hope this brings a smile to your day! Let me know if you have any more questions.
To solve the given differential equation:
dy/dx = y^2/x^3
We will use the method of separation of variables.
Step 1: Separate the variables.
dy/y^2 = dx/x^3
Step 2: Integrate both sides.
∫(dy/y^2) = ∫(dx/x^3)
To integrate the left side, we can rewrite it as:
∫(1/y^2)dy
Using the power rule of integration, the integral becomes:
-1/y = -1/y + C1
To integrate the right side, we can rewrite it as:
∫(1/x^3)dx
Using the power rule of integration, the integral becomes:
-1/2x^2 = -1/2x^2 + C2
Step 3: Simplify the equation.
-1/y = -1/y + C1
-1/y + 1/y = C1
Simplifying further:
1/y = C1
-1/2x^2 = -1/2x^2 + C2
Step 4: Solve for y.
From the equation 1/y = C1, we can rearrange it as:
y = 1/C1
Since we have a condition y(1) = 1, we can substitute it and find the value of the constant C1.
1 = 1/C1
C1 = 1
Therefore, y = 1/C1 = 1/1 = 1.
Hence, the solution to the differential equation is y = 1.
To solve the given differential equation, we'll use the method of separation of variables.
Step 1: Rearrange the equation
Start by rearranging the given differential equation:
dy/dx = y^2/x^3
Cross multiply to get:
xdy = y^2dx/x^3
Step 2: Separate the variables
Next, separate the variables by bringing all terms involving 'y' to one side and all terms involving 'x' to the other side. Divide both sides by y^2 and multiply by x^3:
xdy/y^2 = dx
Step 3: Integrate both sides
Now, integrate both sides of the equation with respect to their respective variables.
∫ xdy/y^2 = ∫ dx
The integral of xdy/y^2 can be evaluated as -1/y + C1, where C1 is the constant of integration.
The integral of dx is simply x + C2, where C2 is another constant of integration.
Step 4: Combine the integrals and solve for y
Combining the integrals from the previous step, we have:
-1/y + C1 = x + C2
Rearranging the equation, we get:
-1/y = x + C - C2, where C = C1 - C2
Now, multiply both sides by -1:
1/y = -x - C + C2
Finally, taking the reciprocal of both sides gives us the solution for y in terms of x:
y = 1 / (-x - C + C2)
Step 5: Applying the initial condition
To find the specific solution for y, we need to apply the initial condition y(1) = 1 to determine the value of the constant C.
When x = 1, y = 1. Substituting these values into the solution equation, we get:
1 = 1 / (-1 - C + C2)
To solve for C, rearrange the equation to isolate C:
1 + C - C2 = -1
C - C2 = -2
C = C2 - 2
Hence, the solution to the given differential equation with the initial condition is:
y = 1 / (-x + C2 - 2)
where C2 is any constant.