Question

Given that y 1​ (x)=e²ˣ is a solution of xy ′′ −(4x+1)y ′ +(4x+2)y=0 Find a second solution, y₂ (x), using reduction of order, and verify that y(x)=c₁​ y₁ (x)+ c₂ y₂​ (x) is the general solution of the ODE for x>0.

Answer 1

The second solution to the differential equation can be found using reduction of order, starting with the known solution y1(x) = e^2x. The function v(x) is determined through substitution and simplification, and is used to find y2(x), then the general solution is verified.

Step-by-step explanation:

To find a second solution, y2(x), to the differential equation using reduction of order, we already have a first solution y1(x) = e2x. We assume a solution of the form y2 = v(x)y1(x), where v(x) is an unknown function to be determined. After substituting y2 and its derivatives into the original differential equation, we can simplify and solve for v(x), typically resulting in an equation involving the integral of e-2x or similar expressions.

Once we've calculated v(x), we can multiply it by y1 to get y2. The general solution y(x) = c1 y1(x) + c2 y2(x), where c1 and c2 are constants, will satisfy the given differential equation for x > 0, as long as y1 and y2 are linearly independent, which we verify using the Wronskian.