Question

solve the given equation by the method of undetermined coefficients. y + 3y - 4y = -28 the solution is y = \square.

Explanation:

Step1: Find the complementary solution

The homogeneous equation is \(y'' + 3y' - 4y=0\). The characteristic equation is \(r^{2}+3r - 4 = 0\).
Factor the quadratic equation: \((r + 4)(r-1)=0\).
So, \(r=-4\) and \(r = 1\).
The complementary solution \(y_{c}=C_{1}e^{-4x}+C_{2}e^{x}\).

Step2: Find a particular solution

Since the non - homogeneous term is a constant (\(-28\)), assume a particular solution of the form \(y_{p}=A\).
Differentiate \(y_{p}\): \(y_{p}' = 0\) and \(y_{p}''=0\).
Substitute into the original equation: \(0+3\times0 - 4A=-28\).
Solve for \(A\): \(-4A=-28\), so \(A = 7\).

Step3: Write the general solution

The general solution of the non - homogeneous equation is \(y=y_{c}+y_{p}\).

Answer:

\(y = C_{1}e^{-4x}+C_{2}e^{x}+7\)