Solving Differential Equations with Initial Conditions

This article explains how to solve the differential equation xfrac{dy}{dx} - 2y x^2 - x - 1 under the initial conditions x 1 and y 1/2. Additionally, we also explore the solution of the homogenous part of the differential equation and the use of an integrating factor.

Solution to the Non-Homogeneous Differential Equation

Consider the differential equation:

xfrac{dy}{dx} - 2y x^2 - x - 1

Divide by x:

frac{dy}{dx} - frac{2y}{x}  x - 1 - frac{1}{x}

Identify an integrating factor:

e^{int -frac{2}{x} dx}  x^{-2}

Multiply the differential equation by the integrating factor:

x^{-2}frac{dy}{dx} - 2x^{-3}y  x^{-1} - x^{-2} - x^{-3}

Integrate:

int x^{-2}frac{dy}{dx} dx int (x^{-1} - x^{-2} - x^{-3}) dx

x^{-2}y ln x - frac{1}{x} frac{1}{2x^2} C

y x^2 (ln x - frac{1}{x} frac{1}{2x^2}) Cx^2

y x^2ln x - x frac{1}{2} Cx^2

Apply initial conditions:

When x 1 and y 1/2:

frac{1}{2} 1 - 1 frac{1}{2} C

C 0

y x^2ln x - x frac{1}{2}

When x 1 and y 3 (check if possible):

3 1 - 1 frac{1}{2} C

C frac{5}{2}

y x^2ln x - x frac{1}{2} frac{5}{2}x^2

However, the above solution contradicts the initial conditions. Let's apply the method of finding a particular solution.

Solving the Homogeneous Part

Consider the homogeneous part of the differential equation:

xy' - 2y 0

Solve the characteristic equation:

frac{dy}{y} frac{2}{x} dx

ln y 2ln x C

y Cx^2

Next, find a particular solution using the form:

y_p Ax^2 Bx C

Compute the derivative:

y_p' 2Ax B

Substitute into the original equation:

x(2Ax B) - 2(Ax^2 Bx C) x^2 - x - 1

2Ax^2 Bx - 2Ax^2 - 2Bx - 2C x^2 - x - 1

Bx - 2Bx - 2C x^2 - x - 1

-x - 2C x^2 - x - 1

Compare coefficients: For x^2: 0 1 (No contradiction) For x: -1 -1 (1 1) For constant: -2C -1 (C 1/2) General solution:

y Cx^2 frac{1}{2}x^2 - x frac{1}{2}

Conclusion

The solution to the differential equation xfrac{dy}{dx} - 2y x^2 - x - 1 under the initial conditions x 1 and y 1/2 is:

y frac{1}{2}x^2 - x frac{1}{2}

The integrating factor method and the method of finding the particular solution were applied to solve the differential equation.