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प्रश्न
The general solution of the differential equation ex dy + (y ex + 2x) dx = 0 is ______.
पर्याय
xey + x2 = C
xey + y2 = C
yex + x2 = C
yey + x2 = C
उत्तर
The general solution of the differential equation ex dy + (y ex + 2x) dx = 0 is yex + x2 = C.
Explanation:
The given equation
exdy + (yex + 2x) dx = 0
or `e^x dy/dx + ye^x + 2x = 0`
`dy/dx + 1 * y = (- 2x)/e^x`
Comparing this equation with `dy/dx + Py = Q.`
P = 1, Q = `(- 2x)/e^x`
∴ `I.F. = e^(int 1 dx) = e^x`
Hence, the general solution of the equation
`y * e^x = int (- 2x)/e^x * e^x dx + C`
`y e^x = int - 2 x dx + C`
`y e^x = - 2 x^2/2 + C`
`y e^x = - x^2 + C`
`y e^x + x^2 = C`
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