Question:
The particular integral of $\frac{d^2y}{dx^2}+2y=x^2$is
The particular integral of $\frac{d^2y}{dx^2}+2y=x^2$is
Updated On: Jun 18, 2022
- $x^2 - 1$
- $x^2 + 1$
- $\frac{1}{2}(x^2-1)$
- $\frac{1}{2}(x^2 + 1)$
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The Correct Option is C
Solution and Explanation
$If \, \, \frac{d^2 y}{dx^3} + 2y \, =x^2$
$\Rightarrow \, \, \, (D^2 + 2)y \, = x^2 \, \, \, \, \bigg[D=\frac{d}{dx}\bigg]$
Particular integral (P.I.) $=\frac{1}{D^2 + 2} x^2$
$= \frac{1}{2\bigg(1 + \frac{D^2}{2}\bigg)} . x^2 =\frac{1}{2} \bigg(1 + \frac{D^2}{2}\bigg)^{-1} . (x^2)$
$\because \, \, (1+D)^{-1} \, = 1 - D + D^2 - D^3 + ....$
$\therefore \, \, P.I. =\frac{1}{2} . \bigg[1- \bigg(\frac{D^2}{2}\bigg)+ \bigg(\frac{D^2}{2}\bigg)^2 - ...... \bigg](x^2)$
$\Rightarrow \, \, P.I. \, = \frac{1}{2}. \bigg[x^2 \, - \frac{D^2}{2} (x^2)\bigg]$
$\Rightarrow \, \, \, P.I. \, = \frac{1}{2} . \bigg[x^2 - 1\bigg]$
$\Rightarrow \, \, \, (D^2 + 2)y \, = x^2 \, \, \, \, \bigg[D=\frac{d}{dx}\bigg]$
Particular integral (P.I.) $=\frac{1}{D^2 + 2} x^2$
$= \frac{1}{2\bigg(1 + \frac{D^2}{2}\bigg)} . x^2 =\frac{1}{2} \bigg(1 + \frac{D^2}{2}\bigg)^{-1} . (x^2)$
$\because \, \, (1+D)^{-1} \, = 1 - D + D^2 - D^3 + ....$
$\therefore \, \, P.I. =\frac{1}{2} . \bigg[1- \bigg(\frac{D^2}{2}\bigg)+ \bigg(\frac{D^2}{2}\bigg)^2 - ...... \bigg](x^2)$
$\Rightarrow \, \, P.I. \, = \frac{1}{2}. \bigg[x^2 \, - \frac{D^2}{2} (x^2)\bigg]$
$\Rightarrow \, \, \, P.I. \, = \frac{1}{2} . \bigg[x^2 - 1\bigg]$
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Concepts Used:
Integrals of Some Particular Functions
There are many important integration formulas which are applied to integrate many other standard integrals. In this article, we will take a look at the integrals of these particular functions and see how they are used in several other standard integrals.
Integrals of Some Particular Functions:
- ∫1/(x2 – a2) dx = (1/2a) log|(x – a)/(x + a)| + C
- ∫1/(a2 – x2) dx = (1/2a) log|(a + x)/(a – x)| + C
- ∫1/(x2 + a2) dx = (1/a) tan-1(x/a) + C
- ∫1/√(x2 – a2) dx = log|x + √(x2 – a2)| + C
- ∫1/√(a2 – x2) dx = sin-1(x/a) + C
- ∫1/√(x2 + a2) dx = log|x + √(x2 + a2)| + C
These are tabulated below along with the meaning of each part.
