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In a second order linear homogeneous differential equation of the form:

$$ ay''(t)+by'(t)+cy(t)=0 $$

the general solution is:

$$ c_1y_1(t)+ c_2y_2(t) $$

Here both $y_1(t)$ and $y_2(t)$ are solutions then why both are added to form new solution that is called general solution.

Could any one clarify me the reason for this.

Many books says that they both are fundamental set of solutions and so can be used to form general solution

Also I am clear that the conditions to be met for the solutions to be fundamental set of solutions.

But My confusion is why all these additions of solutions necessary? Why we cannot use only one solution as general solution?

John Rennie
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Manikandan C
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  • Both y1(t) and y2(t) satisfy the differential equation. Further, we cannot obtain y2(t) from y1(t), so the general solution must have y1(t) and y2(t). Whether it contains only either 1 of either y1(t) or y2(t) or a combination of them is not up for us to decide at this juncture: that is enforced by the boundary conditions. – NaOH Nov 14 '16 at 11:18
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    This question belongs on the Math SE, but it will probably (and should be) be closed there as it most certainly has been answered before, and the question shows insufficient prior research. – JamalS Nov 14 '16 at 11:27

1 Answers1

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why we cannot use only one solution as general solution?

Consider, for example, the differential equation for $a = 1, b = 0, c = \omega^2$. The general solution is

$$y(t) = c_1 \cos(\omega t) + c_2 \sin(\omega t)$$

Why are both required for the general solution? Consider the case when $t=0$:

$$y(0) = c_1 \cos(0) + c_2\sin(0) = c_1$$

Clearly, $c_1$ is the initial position $y(0)$. Now take the derivative and evaluate at $t=0$:

$$\dot y(0) =-c_1 \omega\sin(0) + c_2 \omega\cos(0) = c_2 \omega$$

Clearly, $c_2$ is the initial velocity $\dot y(0)$ divided by the angular frequency $\omega$. Thus, we need both solutions in order to satisfy any combination of initial position and initial velocity:

$$y(t) = y(0) \cos(t) + \frac{\dot y(0)}{\omega} \sin(t) $$

Alfred Centauri
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