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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
$\text{x}^3\frac{\text{d}{^2}\text{y}}{\text{dx}^2}=1$ $\text{y}=\text{ax}+\text{b}+\frac{1}{2\text{x}}$
✓

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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