Differential Equations 1 Question 9
9. A solution of the differential equation $\frac{d y}{d x}{ }^{2}-x \frac{d y}{d x}+y=0$ is
(1999, 2M)
(a) $y=2$
(b) $y=2 x$
(c) $y=2 x-4$
(d) $y=2 x^{2}-4$
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Answer:
Correct Answer: 9. (c)
Solution:
- Given differential equation is
$$ \frac{d y}{d x}^{2}-x \frac{d y}{d x}+y=0 $$
(a) $y=2 \Rightarrow \frac{d y}{d x}=0$
On putting in Eq. (i),
$$ 0^{2}-x(0)+y=0 $$
$\Rightarrow y=0$ which is not satisfied.
(b)
$$ y=2 x \quad \Rightarrow \quad \frac{d y}{d x}=2 $$
On putting in Eq. (i),
$$ \begin{array}{rr} & (2)^{2}-x \cdot 2+y=0 \\ & 4-2 x+y=0 \\ \Rightarrow & y=2 x \text { which is not satisfied. } \\ \Rightarrow \quad y=2 x-4 \Rightarrow \frac{d y}{d x}=2 \end{array} $$
On putting in Eq. (i)
$$ \begin{aligned} (2)^{2}-x-2+y & \\ 4-2 x+2 x-4 & =0 \\ y & =2 x-4 \text { is satisfied. } \\ y & =2 x^{2}-4 \\ \frac{d y}{d x} & =4 x \end{aligned} $$
On putting in Eq. (i),
$$ \begin{aligned} & (4 x)^{2}-x \cdot 4 x+y=0 \\ & \Rightarrow \quad y=0 \text { which is not satisfied. } \end{aligned} $$
