| Class XII |
Mathematics |
Mathematics Part-II |
3 |
2 3 degree 2, 1, 0 respectively but F 4s not a homogeneous function. Reprint 2026-27 DIFFERENTIAL EQUATIONS 313 We also observe that F (x, y) =x 2... |
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| Class XII |
Mathematics |
Mathematics Part-II |
3 |
= g(v) – v ... (4) dx Separating the variables in equation (4), we get dv dx g(v)−v = x ... (5) Integrating both sides of equation (5), we get dv 1... |
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| Class XII |
Mathematics |
Mathematics Part-II |
3 |
= x Therefore ∫cosv dv = ∫x dx Reprint 2026-27 318 MATHEMATICS or sin v = log |x| + log |C| or sin v = log |Cx| y Replacing v by , we get x sin ... |
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| Class XII |
Mathematics |
Mathematics Part-II |
3 |
= C ⇒ C = 2 Substituting the value of C in equation (3), we get x y 2e + log |y| = 2 whichistheparticularsolutionofthegivendifferentialequation. Ex... |
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| Class XII |
Mathematics |
Mathematics Part-II |
3 |
2xy dy = 0 2 dy 2 2 5. x = x −2y + xy 6. x dy – y dx = x + y dx dx xcos + ysin y dx= ysin − xcos x dy 7. ... |
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| Class XII |
Mathematics |
Mathematics Part-II |
3 |
of x only, is known as a first order linear differential equation. Some examples of the first order linear differential equation are dy + y dx = si... |
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| Class XII |
Mathematics |
Mathematics Part-II |
3 |
Q.e ∫P xdx + C ∫ which is the general solution of the differential equation. Reprint 2026-27 324 MATHEMATICS Steps involved to solve first or... |
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| Class XII |
Mathematics |
Mathematics Part-II |
3 |
ye = e +C 2 sin x−cosx + Ce x or y = 2 whichisthegeneralsolutionofthegivendifferentialequation. x dy + 2y = x (x≠0) Example15 Findtheg... |
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| Class XII |
Mathematics |
Mathematics Part-II |
3 |
sin x = ∫(2x + x cot x) sin x dx + C Reprint 2026-27 DIFFERENTIAL EQUATIONS 327 or y sin x = ∫2x sin x dx + ∫x cos x dx + C 2 2 or y sin x ... |
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| Class XII |
Mathematics |
Mathematics Part-II |
3 |
, then – x dx = dt or x dx = – dt. −x2 t t 2 Therefore, I =− ∫e dt = −e= – e Substituting the value of I in equation (2), we get −x2 −x2 ye 2 = 2 +... |
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| Class XII |
Mathematics |
Mathematics Part-II |
3 |
15. −3 cot =sin2 ; = 2 when = x dx 2 16. Findtheequationofacurvepassingthroughtheorigingiventhattheslopeofthe tangent to the curve at any point (x,... |
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| Class XII |
Mathematics |
Mathematics Part-II |
3 |
values of dx 2 dx and y in the given differential equation, we get L.H.S. = e [a c −2a2c −b c )s1nbx+(a 2 +2abc −b c )cosb1] 2 2 1 ax −2ae [(bc +ac... |
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| Class XII |
Mathematics |
Mathematics Part-II |
3 |
2 2 x ysin − x cos dy =xy cos + y sin dx xy cos + y2sin dy or dx = ... |
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| Class XII |
Mathematics |
Mathematics Part-II |
3 |
the differential equation (tan y – x) dy = (1 + y ) dx. Solution The given differential equation can be written as −1 dx + x tan y dy 1+ y 2 = 1+ y... |
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| Class XII |
Mathematics |
Mathematics Part-II |
3 |
given function (implicit or explicit)isasolutionofthecorrespondingdifferentialequation. x –x 2 d y dy 2 (i) xy = a e + b e + x : x 2 + 2 − xy + −2 ... |
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| Class XII |
Mathematics |
Mathematics Part-II |
3 |
dx =1(x ≠ 0) . x x dy 11. Find a particular solution of the differential equation dy + ycotx = 4x cosec x dx π . (x ≠ 0), given that y = 0 wh... |
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| Class XII |
Mathematics |
Mathematics Part-II |
3 |
with dy and terms containing x should remain with dx. ® A differential equation which can be expressed in the form dy dx dx = f (x,y) or dy = g(x, ... |
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| Class XII |
Mathematics |
Mathematics Part-II |
3 |
innocent looking differentialequationcanbe.Fromthesecondhalfofthetwentiethcenturyattention hasbeendrawntotheinvestigationofthiscomplicatednatureoft... |
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| Class XII |
Mathematics |
Mathematics Part-II |
4 |
338 MATHEMATICS Chapter 10 VECTOR ALGEBRA ❖ In most sciences one generation tears down what another has built and what one has established another ... |
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| Class XII |
Mathematics |
Mathematics Part-II |
4 |
is called a vector. Notice that a directed line segment is a vector (Fig 10.1(iii)), denoted as or simply as , and read as ‘vector ’ or ‘vector ’. ... |
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| Class XII |
Mathematics |
Mathematics Part-II |
4 |
.Thus,thecoordinatesofthepointPmay r r alsobeexpressedas(lr,mr,nr). Thenumberslr,mrandnr,proportionaltothedirection cosinesarecalledasdirectionrati... |
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| Class XII |
Mathematics |
Mathematics Part-II |
4 |
the vectors are: (i) Collinear (ii) Equal (iii) Coinitial Solution (i) Collinear vectors : . (ii) Equal vectors : (iii) Coinitialvectors: Fig 10.5 ... |
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| Class XII |
Mathematics |
Mathematics Part-II |
4 |
when the sides of a triangle are taken in order, it leads to zero resultantastheinitialandterminalpointsgetcoincided(Fig10.8(iii)). Reprint 2026-27... |
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| Class XII |
Mathematics |
Mathematics Part-II |
4 |
for any vectora, we have = Here, the zero vector is called the additive identity for the vector addition. 10.5 MultiplicationofaVectorbyaScalar Let... |
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| Class XII |
Mathematics |
Mathematics Part-II |
4 |
respectively, and denoted by i, j and kˆ , respectively (Fig10.13). Now, consider the position vector of a point P(x, y, z) Fig 10.13 as in Fig 10.... |
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