Çözüldü Second-Order Linear Differential Equations with Constant Coefficients

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Solution of The 1st Equation:
The auxiliary equation is 7p^2 + 3p + 2 = 0 with the roots p = -3 / 14 ∓ i·(√47) / 14
C1, C2 ∈ R, i = √(-1)
y(x) = [ e^(-3x / 14) ]·{ C1·cos[ (√47)x / 14 ] + C2·sin[ (√47)x / 14 ] }.

Check by WolframAlpha: https://www.wolframalpha.com/input?i=7y''+3y'+2y=0
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Solution of The 2nd Equation:
7y'' + 3y' = -2...(I)
First, the homogeneous solution (y1) with no second part at the other side of the equation sign;
7y''+ 3y' = 0
The auxiliary equation 7p^2 + 3p = 0 ⇒ p1 = 0, p2 = -7 / 3
y1 = C1·e^(-7x / 3) + C2·e^(0·x) = C1·e^(-7x / 3) + C2
Then the solution based on the second part which is a first-degree polynomial function like y2 = Ax
y2' = A
y2'' = 0
Substituting these derivatives in the entire equation (I);
7·0 + 3A = -2 ⇒ A = -2 / 3, so the solution with the second part is
y2 = -2x / 3
General solution is y = y1 + y2 = C1·e^(-7x / 3) + C2 - 2x / 3.

Check by WolframAlpha:
https://www.wolframalpha.com/input?i=7y''+3y'+2=0