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:

  1. 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),

$$ \begin{aligned} & (4 x)^{2}-x \cdot 4 x+y=0 \\ & \Rightarrow \quad y=0 \text { which is not satisfied. } \end{aligned} $$



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