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In each of the following, $Q .1$ to 5 form a differential equation representing the given family of curves by eliminating arbitrary constants $a$ and $b$.
$y=e^{2 x}(a+b x)$
$y=e^{2 x}(a+b x)$
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Verified Answer
$$
\begin{aligned}
&y=e^{2 x}(a+b x) \\
&y^{\prime}=2 y+b e^{2 x} \\
&y^{\prime \prime}=2 y^{\prime}+2 b e^{2 x}
\end{aligned}
$$
$y^n=2 y^{\prime}+2 b e^{2 x}$ Subtracting (iii) from (ii) we get :
$$
2 y^{\prime}-y^{\prime \prime}=4 y-2 y^{\prime} \quad \Rightarrow \quad y^{\prime \prime}-4 y^{\prime}+4 y=0
$$
\begin{aligned}
&y=e^{2 x}(a+b x) \\
&y^{\prime}=2 y+b e^{2 x} \\
&y^{\prime \prime}=2 y^{\prime}+2 b e^{2 x}
\end{aligned}
$$
$y^n=2 y^{\prime}+2 b e^{2 x}$ Subtracting (iii) from (ii) we get :
$$
2 y^{\prime}-y^{\prime \prime}=4 y-2 y^{\prime} \quad \Rightarrow \quad y^{\prime \prime}-4 y^{\prime}+4 y=0
$$
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