📂Odinary Differential Equations

Linear Combination of Solutions to Homogeneous Linear Differential Equations is Also a Solution

Theorem¹

If $y_{1}, y_{2}$ is a solution to $ay^{\prime \prime}+by^\prime +cy=0$, then $d_{1}y_{1} + d_{2}y_{2}$ is also a solution. Here, $d_{1}, d_{2}$ is any constant.

Description

As can be seen in the proof, it also holds for any $n$ order linear homogeneous differential equation.

Proof

Assume that $y_{1}, y_{2}$ is a solution to $ay^{\prime \prime}+by^\prime +cy=0$. Then the following two equations are satisfied.

$$\begin{align*} d_{1} (ay_{1}^{\prime \prime}+by_{1}^\prime + cy_{1} ) &=0 \\ d_{2} (ay_{2}^{\prime \prime}+by_{2}^\prime + cy_{2}) &=0 \end{align*}$$

If substituting $d_{1}y_{1} + d_{2}y_{2}$ into the given differential equation results in $0$, then the proof is done.

$$\begin{align*} &a(d_{1}y_{1}+d_2y_{2})^{\prime \prime}+b(d_{1}y_{1}+d_2y_{2})^\prime +c(d_{1}y_{1}+d_2y_{2}) \\ =&\ ad_{1}y_{1}^{\prime \prime} + ad_2y_{2}^{\prime \prime} + bd_{1}y_{1}^\prime + bd_2y_{2}^\prime + cd_{1}y_{1} + cd_2y_{2} \\ =&\d_{1}\left( ay_{1}^{\prime \prime} + by_{1}^\prime + cy_{1} \right) + d_2\left( ay_{2}^{\prime \prime} + by_{2}^\prime + cy_{2} \right) \\ =&\ 0 \end{align*}$$

By assumption, since both the first and second brackets are $0$, the equation is satisfied. Therefore, if $y_{1}$ and $y_{2}$ are solutions to the given differential equation, then $d_{1}y_{1}+d_2y_{2}$ is also a solution.

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