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The general solution of $\frac{d y}{d x}+y \tan x=2 x+x^2 \tan x$
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Verified Answer
The correct answer is:
$y=x^2+c \cos x$
Given differential equation,
$\because$ The differential equation is in linear form, so Integrating factor (I.F.) $=e^{\int \tan x d x}=\sec x$ So, solution of given differential Eq. (i), is
$$
\begin{aligned}
y(\sec x) & =\int\left(2 x+x^2 \tan x\right) \sec x d x \\
& =\int 2 x \sec x d x+\int x^2 \tan x \sec x d x \\
& =\int 2 x \sec x d x+x^2 \sec x-\int 2 x \sec x d x \\
\Rightarrow \quad y \sec x & =x^2 \sec x+c \\
\Rightarrow \quad y & =x^2+c \cos x .
\end{aligned}
$$
$\because$ The differential equation is in linear form, so Integrating factor (I.F.) $=e^{\int \tan x d x}=\sec x$ So, solution of given differential Eq. (i), is
$$
\begin{aligned}
y(\sec x) & =\int\left(2 x+x^2 \tan x\right) \sec x d x \\
& =\int 2 x \sec x d x+\int x^2 \tan x \sec x d x \\
& =\int 2 x \sec x d x+x^2 \sec x-\int 2 x \sec x d x \\
\Rightarrow \quad y \sec x & =x^2 \sec x+c \\
\Rightarrow \quad y & =x^2+c \cos x .
\end{aligned}
$$
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