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- Homework Statement
- Solve ##P'=QP## where Q and P are ##n \times n\in## matrices over the reals.
- Relevant Equations
- ##P'=QP##.
I have never solved a matrix ODE before, and am wondering if solving it is similar to solving ##y'=ay## where ##a## is a constant and ##y:\mathbb{R} \longrightarrow \mathbb{R}## is a function. The solution is right according to wikipedia, and I am just looking for your inputs. Thanks
$$\begin{align*}
\frac{d}{dt}P(t)&=QP(t)\\
\frac{1}{P(t)}dP(t)&=Qdt\\
\int\frac{1}{P(t)}dP(t)&=\int Qdt\\
\ln{P(t)}&=Qt+C\\
P(t)&=Ce^{Qt}\\
P(0)&=I\\
\Longrightarrow &Ce^0=I\\
\Longrightarrow &C=I\\
P(t)&=e^{Qt}.
\end{align*}$$
Line 2: Separation of variables.
Lines 3 and 4: Integration with respect to ##t##.
Line 5: Rule of exponents.
Lines 6-8: Determination of the matrix ##C##.
$$\begin{align*}
\frac{d}{dt}P(t)&=QP(t)\\
\frac{1}{P(t)}dP(t)&=Qdt\\
\int\frac{1}{P(t)}dP(t)&=\int Qdt\\
\ln{P(t)}&=Qt+C\\
P(t)&=Ce^{Qt}\\
P(0)&=I\\
\Longrightarrow &Ce^0=I\\
\Longrightarrow &C=I\\
P(t)&=e^{Qt}.
\end{align*}$$
Line 2: Separation of variables.
Lines 3 and 4: Integration with respect to ##t##.
Line 5: Rule of exponents.
Lines 6-8: Determination of the matrix ##C##.
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