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Differential Equations — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDifferential Equations1 Mark
MCQ
If $\frac{d y}{d x}=\frac{x+y}{x}, y(1)=1$, then $y=$
  • A
    $x+\ln x$
  • B
    $x^2+x \ln x$
  • C
    $x e^{x-1}$
  • ✓
    $x+x \ln x$
✓

Answer

Correct option: D.
$x+x \ln x$
(d) : It is a homogeneous equation.
Substitute $y=v x \Rightarrow \frac{d y}{d x}=x \frac{d v}{d x}+v$
Now, given equation becomes
$
x \frac{d v}{d x}+v=1+v \Rightarrow d v=\frac{d x}{x}
$
On integrating both sides, we get
$
\begin{array}{l}
v=\ln x+c \Rightarrow \frac{y}{x}=\ln x+c \\
\because \quad y(1)=1 \Rightarrow x=1, y=1 \Rightarrow c=1 \quad \therefore y=x+x \ln x
\end{array}
$

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