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Determine whether the following systems are linear or not, $ \frac{d y(t)}{d t}+t y(t)=x^2(t) $
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Solution:

$ \frac{d\left[a y_1(t)+b y_2(t)\right]}{d t}+t\left[a y_1(t)+b y_2(t)\right]=\left[a x_1(t)+b x_2(t)\right]^2 \ldots $

$ \frac{d y_1(t)}{d t}+t y_1(t)=x_1^2(t)\\ $

$ \frac{d y_2(t)}{d t}+t y_2(t)=x_2^2(t) \ldots\\ $

$ \text { (2) } \times a+(3) \times b\\ $

$ \Rightarrow a \frac{d y_1(t)}{d t}+a t y_1(t)+b \frac{d y_2(t)}{d t}+b t y_2(t)\\ $

$ =a x_1^2(t)+b x_2^2(t) \ldots(4)\\ $

$(1) \neq(4)$

The given system is Non-Linear

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