If y= ( ) prove that x^2 d^2y dx^2 +x dy dx +y=0 - Vidyadip

Question

If $\text{y}=\sin(\log\text{x})$ prove that $\text{x}^2\frac{\text{d}^2\text{y}}{\text{dx}^2}+\text{x}\frac{\text{dy}}{\text{dx}}+\text{y}=0$

✓

Answer

Here,
$\text{y}=\sin(\log\text{x})$
Differentiating w.r.t.x, we get
$\frac{\text{dy}}{\text{dx}}=\frac{\cos(\log\text{x})}{\text{x}}$
Differentiating w.r.t.x, we get
$\frac{\text{d}^2\text{y}}{\text{dx}^2}=\frac{-\sin(\log\text{x})-\cos(\log\text{x})}{\text{x}^2}$
$\Rightarrow\frac{\text{d}^2\text{y}}{\text{dx}^2}=\frac{-\sin(\log\text{x})}{\text{x}^2}-\frac{\cos(\log\text{x})}{\text{x}^2}{}$
$\Rightarrow\frac{\text{d}^2\text{y}}{\text{dx}^2}=\frac{-\text{y}}{\text{x}^2}-\frac{1}{\text{x}}\times\frac{\text{dy}}{\text{dx}}$
$\Rightarrow\text{x}^2\frac{\text{d}^2\text{y}}{\text{dx}^2}+\text{x}\frac{\text{dy}}{\text{dx}}+\text{y}=0$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "3 Marks Question" from CONTINUITY AND DIFFERENTIABILITY→CONTINUITY AND DIFFERENTIABILITY — all question types→MATHS STD 12 Science question papers→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

Show that the relation R in the set A = {1, 2, 3, 4, 5} given by R = {(a, b) : |a – b| is even}, is an equivalence relation. Show that all the elements of {1, 3, 5} are related to each other and all the elements of {2, 4} are related to each other. But no element of {1, 3, 5} is related to any element of {2, 4}.

Evalute the following integrals:
$\int\big\{1+\tan\text{x}\tan(\text{x}+\theta)\big\}\text{dx}$

Prove the following:$\cos^{-1}\Bigg(\frac{12}{13}\Bigg)+\sin^{-1}\Bigg(\frac{56}{65}\Bigg)$.

Evaluate the following integrals:
$\int^\limits6_{-6}\big|\text{x}+2\big|\text{dx}$

X is taking up subjects - Mathematics, Physics and Chemistry in the examination. His probabilities of getting grade A in these subjects are 0.2, 0.3 and 0.5 respectively. Find the probability that he gets,
Grade A in two subject.

Let $A = IR – \{3\}$ and $B = IR – \{1\}$. Consider the function $f: A \rightarrow B$ defined by $\text{f(x)}=\Bigg(\frac{\text{x - 2}}{\text{x - 3}}\Bigg)$. Show that fis one$-$one and onto and hence find $f^{–1}$.

Prove that every identity relation on a set is reflexive, but the converse is not necessarily true.

Sum the following siries:
$\tan^{-1}\frac{1}{3}+\tan^{-1}\frac{2}{9}+\tan^{-1}\frac{4}{33}+...+\tan^{-1}\frac{2^{\text{n}-1}}{1+2^{2\text{n}-1}}$

Differentiate the following functions with respect to x:
$3^{\text{e}^{\text{x}}}$

If $\text{x}=3\sin\text{t}-\sin3\text{t},\text{y}=3\cos3\text{t}-\cos3\text{t}$ find $\frac{\text{dy}}{\text{dx}}\text{ at t}=\frac{\pi}{3}$