Prove that y - A-x - B is the solution of the differential equation d2y-dx2 - 2-xdy-dx - 0-

Given, y=Ax+B dydx= Ax^2 d^2ydx^2=2Ax^3 Given, d^2ydx^2+2xdydx =2Ax^3+2x× Ax^2 =2Ax^3 2Ax^3 =0 Hence proved. ...

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Standard XII

Mathematics

Particular Solution of a Differential Equation

Prove that ...

Question

Prove that $y = \frac{A}{x} + B$ is the solution of the differential equation $\frac{d^{2} y}{d x^{2}} + \frac{2}{x} \frac{d y}{d x} = 0$ .

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Solution

Given,

$y = \frac{A}{x} + B$

$\frac{d y}{d x} = - \frac{A}{x^{2}}$

$\frac{d^{2} y}{d x^{2}} = \frac{2 A}{x^{3}}$

Given,

$\frac{d^{2} y}{d x^{2}} + \frac{2}{x} \frac{d y}{d x}$

$= \frac{2 A}{x^{3}} + \frac{2}{x} \times - \frac{A}{x^{2}}$

$= \frac{2 A}{x^{3}} - \frac{2 A}{x^{3}}$

$= 0$

Hence proved.

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Similar questions

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

is the solution of differential equation

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

Q. Let

y = a cos (log x) + b sin (log x)

is a solution of the differential equation

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

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[AI CBSE 1986]

Which of the following differential equations has y=x as one of its particular solution?
(a) $\frac{d^{2} y}{d x^{2}} - x^{2} \frac{d y}{d x} + x y = x$
(b) $\frac{d^{2} y}{d x^{2}} + x \frac{d y}{d x} + x y = x$
(c) $\frac{d^{2} y}{d x^{2}} - x^{2} \frac{d y}{d x} + x y = 0$
(d) $\frac{d^{2} y}{d x^{2}} + x \frac{d y}{d x} + x y = 0$

Q. Show that the function

y = A x + \frac{B}{x}

is a solution of the differential equation

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

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Prove that y=Ax+B is the solution of the differential equation d2ydx2+2xdydx=0.

Given,y=Ax+Bdydx=−Ax2d2ydx2=2Ax3Given,d2ydx2+2xdydx=2Ax3+2x×−Ax2=2Ax3−2Ax3=0Hence proved.

Prove that $y = \frac{A}{x} + B$ is the solution of the differential equation $\frac{d^{2} y}{d x^{2}} + \frac{2}{x} \frac{d y}{d x} = 0$ .