Question

The differential equation $(3x+4y+1)dx+(4x+5y+1)dy=0$ represents a family of

• A. circles
• B. parabolas
• C. ellipses
• D. hyperbolas

Answer 1

Answer: (D)

The given differential equation is
$(3x+4y+1)dx+(4x+5y+1)dy=0$ ....(i)
Comparing eq. (i) with $Mdx+Ndy=0$,
we get
$M=3x+4y+1$
and $N=4x+5y+1$
Here, $\frac{\partial M}{\partial y}=\frac{\partial N}{\partial x}=4$
Hence, eq. (i) is exact and solution is given by
$\int \left(3x+4y+1\right)dx+\int \left(5y+1\right)dy=C$
$\Rightarrow \frac{3x^2}{2}+4xy+x+\frac{5y^2}{2}+y-C=0$
$\Rightarrow 3x^2+8xy+2x+5y^2+2y-C=0$
$\Rightarrow 3x^2+2.4xy+2x+5y^2+2y+C^{\prime }=0$ ....(ii)
where, $C^{\prime }=-2C$
On comparing eq. (ii) with standard form of conic section
$a{x}^2+2hxy+b{y}^2+2gx+2fy+C=0$
We get,
$a=3,h=4,b=5$
Here, $h^2-ab=16-15=1>0$
Hence, the solution of differential equation represents family of hyperbolas.