Fnd the equation of the curve through the point (-1,2) of the differential equation dy/dx = -2xy/(x^2+1)
the solution to the d.e. is
y = c/(x^2+1)
c/2 = 2
c = 4
y = 4/(x^2+1)
@steve... i got c=2 log 2 .. before putting the pts i got the equation logy+log(1+x^2)=logc
@steve... now i further solved it... thnx.. ur answer is correct :) :)
To find the equation of the curve through the point (-1, 2) of the differential equation dy/dx = -2xy/(x^2+1), we can solve the differential equation using separation of variables.
Step 1: Separate the variables
Start by rearranging the equation to isolate y and x terms on different sides.
(y)/(y^2+1) dy = -2x/(x^2+1) dx
Step 2: Integrate both sides
Now we need to integrate both sides of the equation.
∫ (y)/(y^2+1) dy = ∫ -2x/(x^2+1) dx
On the left side, the integral is a trigonometric substitution. Substitute y = tan(theta), dy = sec^2(theta) d(theta).
∫ (tan(theta))/(tan^2(theta) + 1) sec^2(theta) d(theta) = ∫ -2x/(x^2+1) dx
This becomes:
∫ tan(theta) d(theta) = ∫ -2x/(x^2+1) dx
Using the identity for the integral of tangent, the left side simplifies to:
ln|sec(theta)| = ∫ -2x/(x^2+1) dx
Step 3: Convert back to x and y
We now convert the trigonometric terms back into x and y variables.
Since y = tan(theta), we can find theta using the inverse tangent function:
theta = arctan(y)
Since sec(theta) = √(tan^2(theta) + 1), we have:
sec(theta) = √(y^2 + 1)
So the equation becomes:
ln|√(y^2 + 1)| = ∫ -2x/(x^2+1) dx
Step 4: Evaluate the integral
To find the integral on the right side, we can use the substitution u = x^2 + 1, du = 2x dx.
Substituting this back into the equation, we get:
ln|√(y^2 + 1)| = ∫ -1/u du
Integrating, we have:
ln|√(y^2 + 1)| = -ln|u| + C
Step 5: Solve for y
To solve for y, we can exponentiate both sides of the equation.
e^(ln|√(y^2 + 1)|) = e^(-ln|u| + C)
Simplifying, we have:
√(y^2 + 1) = e^(-ln|u| + C)
Since e^(-ln|u| + C) = 1/(eu), we can rewrite the equation as:
√(y^2 + 1) = 1/(eu)
Step 6: Solve for y^2
To isolate y^2, we square both sides of the equation.
(y^2 + 1) = 1/(eu)^2
Simplifying, we get:
y^2 + 1 = 1/(eu)^2
Step 7: Substitute the initial condition
Now we substitute the point (-1, 2) into the equation and solve for the constant C.
Plugging in x = -1 and y = 2, we have:
(2)^2 + 1 = 1/(e(-1)^2)^2
Simplifying the right side, we get:
5 = 1/(e^2)
Solving for the constant C, we have:
C = ln(1/√5)
Step 8: Final equation
Replacing the constant, the equation becomes:
y^2 + 1 = 1/(e^u)^2
Therefore, the equation of the curve through the point (-1, 2) of the differential equation dy/dx = -2xy/(x^2+1) is:
y^2 + 1 = 1/(e^((x^2+1)/2))^2