Verify that x y-a ex-b e-x-x2 is a solution of the differential equation x d 2 y d x 2-2 d y d x-x y-x 2-2-0-

We #160;have,xy=aex+be x+x2Differentiating #160;with #160;respect #160;to #160;x #160;on #160;both #160;sides, #160;we #160;get #8658;xdyd ...

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Standard XII

Mathematics

Ordinary Differential Equations

Verify that x...

Question

Verify that xy = a e^x + b e^−x + x² is a solution of the differential equation $x \frac{d^{2} y}{d x^{2}} + 2 \frac{d y}{d x} - x y + x^{2} - 2 = 0$ .

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Solution

^{$We have, x y = a e^{x} + b e^{- x} + x^{2} Differentiating with respect to x on both sides, we get \Rightarrow x \frac{d y}{d x} + y = a e^{x} - b e^{- x} + 2 x Again differentiating with respect to x on both sides, we get \Rightarrow x \frac{d^{2} y}{d x^{2}} + \frac{d y}{d x} + \frac{d y}{d x} = a e^{x} + b e^{- x} + 2 \Rightarrow x \frac{d^{2} y}{d x^{2}} + 2 \frac{d y}{d x} = x y - x^{2} + 2 [∵ x y = a e^{x} + b e^{- x} + x^{2}] \Rightarrow x \frac{d^{2} y}{d x^{2}} + 2 \frac{d y}{d x} - x y + x^{2} - 2 = 0$}
Thus, xy = a e^x + b e^−x + x² is the solution of the given differential equation.

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Similar questions

Q. For each of the exercises given below, verify that the given function (implicit or explicit) is a solution of the corresponding differential equation.

y = a e^{x} + b e^{- x} + x^{2}

x \frac{d^{2} y}{d x^{2}} + 2 \frac{d y}{d x} - x y + x^{2} - 2 = 0

For the question given below verify that the given function (implicit or explicit) is a solution of the corresponding differential equation.

$y = a e^{x} + b e^{- x} + x^{2}; x \frac{d^{2} y}{d x^{2}} + 2 \frac{d y}{d x} - x y + x^{2} - 2 = 0$

For each of the exercises given below, verify that the given function (implicit or explicit) is a solution of the corresponding differential equation.

(i)

(ii)

(iii)

(iv)

Q. Show that y² − x² − xy = a is a solution of the differential equation

(x - 2 y) \frac{d y}{d x} + 2 x + y = 0

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Verify that xy = a ex + b e−x + x2 is a solution of the differential equation xd2ydx2+2dydx-xy+x2-2=0.

Verify that xy = a e^x + b e^−x + x² is a solution of the differential equation $x \frac{d^{2} y}{d x^{2}} + 2 \frac{d y}{d x} - x y + x^{2} - 2 = 0$ .