Question

Find the general solution of the linear equation of the first order \(y^{\prime}+p(x) y=q(x)\) if one particular solution, \(\mathrm{y}_{1}(\mathrm{x})\), is known.

Short answer

Question: Find the general solution of the given first-order linear differential equation, given a particular solution \(y_{1}(x)\). Answer: The general solution of the given first-order linear differential equation is: \(y(x) = \frac{1}{μ(x)}(\int μ(x)q(x) dx + C)\) where \(μ(x) = e^{\int p(x) dx}\) is the integrating factor, q(x) is the inhomogeneous term in the equation, p(x) is the coefficient of the y term, and C is a constant determined by the known particular solution \(y_{1}(x)\).

Step-by-step solution

Identify the given equation and known solution

The given first-order linear differential equation is: \(y^{\prime} + p(x)y = q(x)\) And the known particular solution is: \(y_{1}(x)\)

Determine the integrating factor

To find the integrating factor, we will use the formula: \(μ(x) = e^{\int p(x) dx}\)

Multiply the given equation by the integrating factor

Now, we will multiply both sides of the given equation by the integrating factor: \(μ(x)(y^{\prime} + p(x)y) = μ(x)q(x)\) This will result in a new equation that is easier to integrate.

Simplify the left side of the equation

The left side of the equation can be simplified by recognizing it as the derivative of the product of a function and its integrating factor, which can be written as: \(\frac{d}{dx}(y(x)·μ(x))= μ(x)q(x)\)

Integrate both sides of the equation

Now, we will integrate both sides of the equation with respect to x: \(\int\frac{d}{dx}(y(x)·μ(x)) dx = \int μ(x)q(x) dx\) Which simplifies to: \(y(x)·μ(x) = \int μ(x)q(x) dx + C\)

Solve for the general solution y(x)

To find the general solution y(x), we will multiply both sides of the equation with the inverse of the integrating factor \(μ^{-1}(x)\): \(y(x) = \frac{1}{μ(x)}(\int μ(x)q(x) dx + C)\)

Substitute the known particular solution y₁(x)

Now that we have the general solution, we can substitute the known particular solution and find the corresponding constant C. Plug in y₁(x) into the general solution and compare it to the given particular solution y₁(x) to find the value of C.

Write the final general solution

With the value of C found, substitute it back into the general solution to obtain the final general solution of the given first-order linear differential equation. This general solution will have the form: \(y(x) = \frac{1}{μ(x)}(\int μ(x)q(x) dx + C)\)