1.

If an integral curve of the differential equation \(\left( {y - x} \right)\frac{{dy}}{{dx}} = 1\) passes through (0, 0) and (α, 1), then α is equal to1. 2 – e-12. 1 – e-13. e-14. 1 + e

Answer» Correct Answer - Option 3 : e-1

The differential equation is:

\(\left( {y - x} \right)\frac{{dy}}{{dx}} = 1\)

The above equation can be written as:

\(\frac{{dx}}{{dy}} + x = y\)

\(I.F. = {e^{\smallint dy}} = {e^y}\)

Then solution is

x.ey = ∫ yey dy + c

x.ey = (y - 1) ey + c

so x = (y - 1) + ce-y

Curve passing through (0, 0) so,

0 = 0 – 1 + c ⇒ c = 1

Now, x = (y - 1) + e-y

If curve passes through (α, 1), then

\(a = \left( {1 - 1} \right) + {e^{ - 1}} = \frac{1}{e}\)


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