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Consider the differential equation $\frac{d y}{d x}=\frac{1}{a x+4 y+7}$ and the following statements
A. The given differential equation is linear in $x$.
$B$. The given differential equation is not linear in $y$.
C. The given differential equation is linear in $y$.
D. $e^{a x}$ is the integrating factor of the given differential equation.
Which one of the following options is true?
Options:
A. The given differential equation is linear in $x$.
$B$. The given differential equation is not linear in $y$.
C. The given differential equation is linear in $y$.
D. $e^{a x}$ is the integrating factor of the given differential equation.
Which one of the following options is true?
Solution:
1768 Upvotes
Verified Answer
The correct answer is:
Only B and A are true
We have, $\frac{d y}{d x}=\frac{1}{a x+4 y+7}$
$\Rightarrow \quad \frac{d x}{d y}=a x+4 y+7 \Rightarrow \frac{d x}{d y}-a x=4 y+7$
Which is form of $\frac{d x}{d y}+P x=Q$
$\therefore$ Differential equation is linear in $x$ but not linear in $y$
$\mathrm{IF}=e^{\int a d y}=e^{a y}$
$\therefore A$ and $B$ are true.
$\Rightarrow \quad \frac{d x}{d y}=a x+4 y+7 \Rightarrow \frac{d x}{d y}-a x=4 y+7$
Which is form of $\frac{d x}{d y}+P x=Q$
$\therefore$ Differential equation is linear in $x$ but not linear in $y$
$\mathrm{IF}=e^{\int a d y}=e^{a y}$
$\therefore A$ and $B$ are true.
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