Question

Find all solutions to below and explain why it must be constant coefficient.

Answer 1

The solution to the differential equation will be expressed as $y_c=Ce^{2x}$

Given the differential equation;

$\frac{d^2y}{dx^2}+2y \frac{dy}{dx} =0$

Since the right-hand side of the homogenous equation is zero, the differential equation is a homogenous equation

Let $D=\frac{dy}{dx}$

The differential equation becomes;

$D^2y-2Dy=0\\(D^2-2D)y=0$

Factorizing the auxiliary equation m² - 2m = 0

m² = 2m

m = 2

Since the auxiliary solution is real and distinct, the solution to the differential equation will be expressed as  $y_c=C_1e^{m_1x} + C_1e^{m_2x}$

Given that m =2, the solution will be expressed as $y_c=Ce^{2x}$

b) Most linear differential equations have a constant coefficient because they are mostly solved using the quadratic method which is also called the method of quadrature. This means that the solution maybe expressed in integral form.

Learn more here: https://brainly.com/question/13512633