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Question: If $y=\left(\cot ^{-1}\right)^{2}$, then show that $\left(x^{2}+1\right)^{2} \frac{d^{2} y}{d x^{2}}+2 x\left(x^{2}+1\right) \frac{d y}{d x}=2$ Solution:...
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Question: If $y=x^{x}$, prove that $\frac{d^{2} y}{d x^{2}}-\frac{1}{y}\left(\frac{d y}{d x}\right)^{2}-\frac{y}{x}=0$ Solution:...
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Question: If $x=a\left(\cos \theta+\log \tan \frac{\theta}{2}\right)$ and $y=a \sin \theta$, show that the value of $\frac{d^{2} y}{d x^{2}}$ at $\theta=\frac{\pi}{4}$ is $\frac{2 \sqrt{2}}{a}$. Solution:...
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Question: If $y=e^{x} \sin x$, prove that $\frac{d^{2} y}{d x^{2}}-2 \frac{d y}{d x}+2 y=0$ Solution:...
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Question: If $y=\frac{\sin ^{-1} x}{\sqrt{1-x^{2}}}$, show that $\left(1-x^{2}\right) \frac{d^{2} y}{d x^{2}}-3 x \frac{d y}{d x}-y=0$ Solution:...
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Question: If $y=\sin (\log x)$, prove that $x^{2} \frac{d^{2} y}{d x^{2}}+x \frac{d y}{d x}+y=0$ Solution:...
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Question: If $x=a(\theta-\sin \theta)$ and $y=a(1-\cos \theta)$, find $\frac{d^{2} y}{d x^{2}}$ at $\theta=\pi$ Solution:...
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Question: If $x=a t^{2}$ and $y=2$ at, find $\frac{d^{2} y}{d x^{2}}$ at $t=2$ Solution:...
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Question: If $y=e^{a \cos -1 x},-1 \leq x \leq 1$, show that $\left(1-x^{2}\right) \frac{d^{2} y}{d x^{2}}-x \frac{d y}{d x}-a^{2} y=0$ Solution: $\Rightarrow\left(1-x^{2}\right) \frac{d^{2} y}{d x^{2}}-a^{2} y-x \frac{d y}{d x}=0$ Hence Proved...
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Question: If $y=e^{a x} \cos b x$, show that $\frac{d^{2} y}{d x^{2}}-2 a \frac{d y}{d x}+\left(a^{2}+b^{2}\right)=0$ Solution:...
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Question: If $y=\frac{\log x}{x}$, show that $\frac{d^{2} y}{d x^{2}}=\frac{(2 \log x-3)}{x^{3}}$ Solution:...
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Question: If $y=e^{\tan }$, prove that $\left(\cos ^{2} x\right) \frac{d^{2} y}{d x^{2}}-(1+\sin 2 x) \cdot \frac{d y}{d x}=0$ Solution:...
Read More →In the circuit shown in figure (32-E13),
Question: In the circuit shown in figure (32-E13), $\varepsilon_{1}=3 \mathrm{~V}, \varepsilon_{2}=2 \mathrm{~V}, \varepsilon_{3}=1 \mathrm{~V}$ and $\mathrm{r}_{1}=\mathrm{r}_{2}=r_{3}=1 \Omega$. Find the potential difference between the points $A$ and $B$ and the current through each branch. Solution:...
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Question: Find the second derivative of $\sin 3 x \cos 5 x$. Solution: $y=\frac{1}{2}[\sin (5 x+3 x)+\sin (5 x-3 x)]$...
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Question: Find the second derivative of $e^{3 x} \sin 4 x$. Solution:...
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Question: If $y=a \cos (\log x)+b \sin (\log x)$, prove that $x^{2} y_{2}+x y_{1}+y=0$ Solution:...
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Question: If $y=\sin (\sin x)$, prove that $\frac{d^{2} y}{d x^{2}}+(\tan ) \frac{d y}{d x}+y \cos ^{2} x=0$ Solution: Hence Proved...
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Question: Find the potential difference $\left(\mathrm{V}_{\mathrm{a}}-\mathrm{V}_{\mathrm{b}}\right)$ in the circuits shown in figure (32-E12). Solution: It is similar to above circuit in (a). Hence same answer....
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Question: If $y=\tan ^{-1}$, show that $\left(1+x^{2}\right) \frac{d^{2} y}{d x^{2}}+2 x \frac{d y}{d x}=0$ Solution:...
Read More →Each of the resistors shown in figure (32-E11) has a resistance of
Question: Each of the resistors shown in figure (32-E11) has a resistance of $10 \Omega$ and each of the batteries has an emf of $10 \mathrm{~V}$. Find the currents through the resistors a and $b$ in the two circuits. Solution:...
Read More →Figure (32-E10) shows a part of an electric circuit.
Question: Figure (32-E10) shows a part of an electric circuit. The potentials at the points $a$, $b$ and $c$ are $30 \mathrm{~V}, 12 \mathrm{~V}$ and $2 \mathrm{~V}$ respectively. Find the currents through the three resistors. Solution:...
Read More →Find the currents through the three resistors shown in figure (32-E9).
Question: Find the currents through the three resistors shown in figure (32-E9). Solution:...
Read More →Find the current measured by the ammeter in the circuit
Question: Find the current measured by the ammeter in the circuit shown in the figure (32-E18). Solution:...
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Question: If $y=(\operatorname{cosec} x+\cot x)$, prove that $(\sin x) \frac{d^{2} y}{d x^{2}}-y^{2}=0$ Solution:...
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