(1+x^4)dy+x(1+4y^2)dx=0 Y(1)=0

4 min read Jun 16, 2024

Solving the Differential Equation: (1+x^4)dy + x(1+4y^2)dx = 0 with y(1) = 0

This article will guide you through the solution of the given differential equation with the initial condition. We will use the method of separation of variables to find the general solution, then apply the initial condition to find the particular solution.

Step 1: Separating the Variables

Begin by rearranging the equation to separate the variables:

(1+x^4)dy = -x(1+4y^2)dx

Next, divide both sides by (1+4y^2) and (1+x^4):

dy/(1+4y^2) = -xdx/(1+x^4)

Now, we have successfully separated the variables, with y on the left-hand side and x on the right-hand side.

Step 2: Integrating Both Sides

Integrate both sides of the equation:

∫dy/(1+4y^2) = -∫xdx/(1+x^4)

The integral on the left-hand side can be solved by using the substitution u = 2y, leading to:

(1/2)∫du/(1+u^2) = -∫xdx/(1+x^4)

Solving these integrals gives us:

(1/2)arctan(2y) = -(1/2)arctan(x^2) + C

Where C is the constant of integration.

Step 3: Solving for y

Multiply both sides by 2 and isolate y:

arctan(2y) = -arctan(x^2) + 2C

Take the tangent of both sides:

2y = tan(-arctan(x^2) + 2C)

Using the tangent addition formula, we can simplify this to:

2y = (-tan(arctan(x^2)) + tan(2C))/(1 + tan(arctan(x^2))tan(2C))

Since tan(arctan(x^2)) = x^2, we obtain:

2y = (-x^2 + tan(2C))/(1 + x^2tan(2C))

Finally, solve for y:

y = (-x^2 + tan(2C))/(2(1 + x^2tan(2C)))

This is the general solution to the differential equation.

Step 4: Applying the Initial Condition

We are given the initial condition y(1) = 0. Substitute these values into the general solution:

0 = (-1 + tan(2C))/(2(1 + tan(2C)))

This implies that:

-1 + tan(2C) = 0

Therefore:

tan(2C) = 1

Solving for C, we find:

C = π/8

Step 5: The Particular Solution

Substitute the value of C back into the general solution to obtain the particular solution:

y = (-x^2 + tan(π/4))/(2(1 + x^2tan(π/4)))

Simplifying further, we get:

y = (-x^2 + 1)/(2(1 + x^2))

This is the particular solution to the given differential equation with the initial condition y(1) = 0.

Conclusion

By applying the method of separation of variables and using the given initial condition, we successfully derived the particular solution to the differential equation (1+x^4)dy + x(1+4y^2)dx = 0 with y(1) = 0. The solution is y = (-x^2 + 1)/(2(1 + x^2)).