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DIFFERENTIAL EQUATIONS — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDIFFERENTIAL EQUATIONS3 Marks
Question
For the following differntial equations verify that the accompanying function is a solution:
Differential equation Function
$\text{x}^3\frac{\text{d}{^2}\text{y}}{\text{dx}^2}=1$ $\text{y}=\text{ax}+\text{b}+\frac{1}{2\text{x}}$
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Answer

We have

$\text{y}=\text{ax}+\text{b}+\frac{1}{2\text{x}}\ ...(1)$

Differentiating both sides of (1) with respect to x, we get

$\frac{\text{dy}}{\text{dx}}=\text{a}-\frac{1}{2\text{x}^2}\ ...(2)$

Now, differentiating both sides of (2) with respect to x, we get

$\Rightarrow\frac{\text{d}^2\text{y}}{\text{dx}^2}=\Big(-\frac{1}{2}\Big)\times\Big(\frac{-2}{\text{x}^3}\Big)$

$\Rightarrow\frac{\text{d}^2\text{y}}{\text{dx}^2}=\frac{1}{\text{x}^3}$

$\Rightarrow\text{x}^3\frac{\text{d}^2\text{y}}{\text{dx}^2}=1$

Hence, the given function is the solution to the given differential equation.

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