Question

Prove that the function,

y(x)=1/e^∫ p(x) d x[e^∫ p(x) d x· q(x)+C]
is a solution to the general linear differential equation,
yʹ+p(x) y=q(x)

Answer 1

Proving that a given function is a solution to a differential equation involves differentiating the function, substituting back into the original equation, and confirming that the resulting expression matches the original equation.

Step-by-step explanation:

To prove that the function y(x) = \frac{1}{e^{\int p(x) dx}} [e^{\int p(x) dx} \cdot q(x) + C] is a solution to the general linear differential equation y' + p(x) y = q(x), we need to follow several steps involving differentiation and substitution.

Step 1: Differentiate y(x)

First, differentiate y(x) with respect to x. Due to the product rule and the chain rule in differentiation, we will obtain the first derivative y' of y that will contain terms with p(x) and q(x).

Step 2: Substitute into the original equation

Next, substitute y and y' back into the linear differential equation y' + p(x)y = q(x). The terms involving e^{\int p(x) dx} and p(x) are expected to cancel appropriately, leaving us with the original q(x) term.

Step 3: Verify the solution

Finally, verify by simplification that the resulting expression matches the right-hand side of the original differential equation, affirming that y(x) is indeed a solution to y' + p(x)y = q(x).