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How To Solve Second Order Linear Homogeneous Differential Equation
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To solve any Second Order Linear Homogeneous Differential Equation, first this you need to do, is to transform the equation in to an auxiliary or characteristics equation in the form: ar²+be+c=0
We have already seen how to do that in our previous lesson.
The next move is to solve for r which are the roots of the equation (r intercept).
Then determine the nature of the roots and substitute in to the following equations, depending on the nature of roots.
y=C₁eʳ¹ˣ+C₂eʳ²ˣ. when you obtain real and distinct roots
y=(C₁+C₂x)eʳˣ when you Obtain real and equal roots.
and finally, if you Obtain a complex solution in the form: r= m+si or r = m-si
where i is imaginary number, and m and s are real numbers, then
y=eᵐˣ[C₁cos(st)+C₂sin(st)]
We have already seen how to do that in our previous lesson.
The next move is to solve for r which are the roots of the equation (r intercept).
Then determine the nature of the roots and substitute in to the following equations, depending on the nature of roots.
y=C₁eʳ¹ˣ+C₂eʳ²ˣ. when you obtain real and distinct roots
y=(C₁+C₂x)eʳˣ when you Obtain real and equal roots.
and finally, if you Obtain a complex solution in the form: r= m+si or r = m-si
where i is imaginary number, and m and s are real numbers, then
y=eᵐˣ[C₁cos(st)+C₂sin(st)]
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