By H.S. Gangwar, Dr. Prabhakar Gupta

ISBN-10: 8122424759

ISBN-13: 9788122424751

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**Extra resources for A Textbook of Engineering Mathematics: v. 1**

**Sample text**

Find the first order derivatives of LMAns. N (i) u = xxy. (ii) u = log 3. If u = sin–1 FH x + x − y IK LMAns. ∂u = 1 ∂u = − y ex MN ∂x x − y ∂y F x − y I , show that ∂u = – y ∂u . GH x + y JK ∂x x ∂y 2 2 2 4. If u = ex (x cos y – y sin y), prove that 5. If u = a log (x2 + y2) + b tan–1 x2 x y2 y z2 z FG y IJ , prove that H xK 1 1 1 , prove that g 2 2 − y2 ∂ 2u ∂ 2u + = 0. ∂x 2 ∂y 2 6. If z = tan (y – ax) + (y + ax)3/2, prove that 7. If u = b OP Q IK OP PQ ∂u ∂u = x xy y log x + y ; = x xy + 1 log x ∂x ∂y ∂ 2u ∂ 2u + = 0.

V) Replace n by (n – 2) in relation (v), we get yn (0) = {(n – 2)2 – a2}yn–2 (0) Again replace n by (n – 4) in equation (v) and putting yn–2 (0) in above relation, we get yn (0) = {(n – 2)2 – a2} {(n – 4)2 – a2}yn–4 (0) Case I. When n is odd: yn (0) = {(n – 2)2 – a2} {(n – 4)2 – a2} ..... (vi) [The last term in (vi) obtain by putting n = 1 in equation (v)] Putting x = 0, in equation (i), we get y1 (0) = cos(a sin–1 0). a = cos 0 . a ⇒ y1 (0) = a yn (0) = {(n – 2)2 – a2} {(n – 4)2 – a2} ..... {12 – a2}.

The given function is homogeneous function of degree 4. r. r. (iii) 47 DIFFERENTIAL CALCULUS-I Adding (ii) and (iii), we get or F GH I JK x2 2 ∂u ∂u ∂ 2u ∂ 2u 2∂ u +y = 3 x xy y 2 + + 2 2 ∂x ∂y ∂x∂y ∂x ∂y x2 2 ∂ 2u ∂ 2u 2∂ u = 3 × 4u = 12u. Hence proved. 5 Verify Euler’s theorem 1. 2 2 x −y . 1 4 4. x 3 y − 3 tan–1 FG y IJ. H xK 2. Fx GH 5. b x + yg . 1 4 1 + y4 I Fx JK GH 1 5 1 I JK + y5 . xy 3. cos–1 6. cos–1 F xI GH y JK F xI GH y JK + cot–1 + cot–1 FG y IJ. H xK FG y IJ . H xK F x + y I , show that x ∂u + y ∂u = 3.

### A Textbook of Engineering Mathematics: v. 1 by H.S. Gangwar, Dr. Prabhakar Gupta

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