conversations listlengths 2 2 |
|---|
[
{
"from": "user",
"value": "Show that (0, 0) is a stable critical point of the nonlinear plane autonomous system\\n\\nx' = -y - x\\sqrt{x^2 + y^2}\\ny' = x - y\\sqrt{x^2 + y^2}\\n\\nconsidered in Example 6 of Section 10.1."
},
{
"from": "assistant",
"value": "<think>\nOkay, so I need to show tha... |
[
{
"from": "user",
"value": "Aniline and nitrosobenzene, in the presence of a base, react to give azobenzene.\\n\\n\\[\\n\\text{PhNH}_2 + \\text{PhNO} \\rightarrow \\text{PhN═NPh} + \\text{H}_2\\text{O}\\n\\]\\n\\nFrom the following observations suggest a mechanism for this reaction.\\n(a) The rate of reacti... |
[
{
"from": "user",
"value": "A spherical balloon, having a charge uniformly distributed over its surface, is inflated so that at a certain instant, every point has a radial velocity $u$. It can be shown that this radial movement of charges produces no magnetic field at any point. Discuss, either in terms of ... |
[
{
"from": "user",
"value": "Give an example of a nonlinear map \\( F: V \\to U \\) such that \\( F^{-1}(0) = \\{0\\} \\) but \\( F \\) is not one-to-one."
},
{
"from": "assistant",
"value": "<think>\nOkay, so I need to find a nonlinear map F from a vector space V to another vector space U where ... |
[
{
"from": "user",
"value": "Solve \\(y' - 5y = 0\\)."
},
{
"from": "assistant",
"value": "<think>\nOkay, so I need to solve the differential equation \\( y' - 5y = 0 \\). Hmm, let's see. I remember that this is a first-order linear differential equation. The standard form for such equations is \... |
[
{
"from": "user",
"value": "In the Bohr model, the electron of a hydrogen atom moves in a circular orbit of radius 5.3 × 10^{-11} m with a speed of 2.2 × 10^{6} m/s. Determine its frequency f and the current I in the orbit."
},
{
"from": "assistant",
"value": "<think>\nOkay, so I need to find th... |
[
{
"from": "user",
"value": "A 2-\\(\\mu\\)F capacitor, with initial charge \\( Q_0 = 100 \\, \\mu \\text{C} \\), is connected across a 100-\\(\\Omega\\) resistor at \\( t = 0 \\). Calculate the time in which the transient voltage across the resistor drops from 40 to 10 volts."
},
{
"from": "assistan... |
[
{
"from": "user",
"value": "Show that the solution for the \\( R \\)-equation in Problem 7.1 can be written as\\n\\n\\[\\nR = Ar^n + \\frac{B}{r^{n+1}}\\n\\]\\n\\nwhere \\( \\lambda^2 = -n(n + 1) \\).\\n\\nThe \\( R \\)-equation of Problem 7.1 is\\n\\n\\[\\nr^2 \\frac{d^2R}{dr^2} + 2r \\frac{dR}{dr} + \\lam... |
[
{
"from": "user",
"value": "An object is placed in front of a concave mirror having a radius of curvature of 0.3 m. If you want to produce first a real image and then a virtual image 3 times as high as the object, find the object distance required in each case."
},
{
"from": "assistant",
"value"... |
[
{
"from": "user",
"value": "What are the key gluconeogenic enzymes?"
},
{
"from": "assistant",
"value": "<think>\nOkay, so I need to figure out the key enzymes involved in gluconeogenesis. Let me start by recalling what gluconeogenesis is. From what I remember, it's the process by which the body... |
[
{
"from": "user",
"value": "What is the meaning of each of the following: (a) ten squared, (b) ten cubed, (c) ten to the fourth power?"
},
{
"from": "assistant",
"value": "<think>\nOkay, so I need to figure out what each of these terms means: ten squared, ten cubed, and ten to the fourth power. ... |
[
{
"from": "user",
"value": "Toluene undergoes oxidation to give (a) Benzyl alcohol (b) Quinone (c) Benzaldehyde (d) Benzoic acid"
},
{
"from": "assistant",
"value": "<think>\nOkay, so I need to figure out what happens when toluene undergoes oxidation. The options are benzyl alcohol, quinone, ben... |
[
{
"from": "user",
"value": "Using the shell model, find the spin–isospin dependence of the wavefunctions for the tritium (3H) and helium (3He) nuclear ground-states."
},
{
"from": "assistant",
"value": "<think>\nOkay, so I need to figure out the spin-isospin dependence of the wavefunctions for t... |
End of preview. Expand in Data Studio
README.md exists but content is empty.
- Downloads last month
- 26