By H.S. Gangwar, Dr. Prabhakar Gupta

ISBN-10: 8122424759

ISBN-13: 9788122424751

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**Additional info for A Textbook of Engineering Mathematics: v. 1**

**Sample text**

Y. t. t. x. (B) 2 Thus, from (A) and (B), we find ∂ 2u ∂ 2u = . ∂y∂x ∂x∂y Hence proved. FG y IJ , verify that ∂ f H xK ∂y∂x F yI f = tan G J H xK 2 Example 2. If f = tan–1 Sol. We have = ∂2 f . (iv) 2 Differentiating (iii) partially with respect to x, we get F GH ∂2 f x ∂ = 2 ∂x∂y ∂x x + y 2 = Example 3. If u(x + y) = x Sol. (v) F ∂u ∂u I + y , prove that G − J H ∂x ∂y K F1 − ∂u − ∂u I . GH ∂x ∂y JK 2 =4 x2 + y2 x+y bx + yga2xf − ex + y ja1f = x + 2xy − y b x + yg bx + y g bx + ygb2yg − ex + y ja1f = y + 2xy − x bx + yg bx + yg 2 = ∂u ∂y = ∂u ∂u + ∂x ∂y = b x + yg = 1− = ex 2 2 2 2 2 2 = 2 2 2 2 2 2 2 2 4xy 2 4xy ( x + y) 2 = (x − y) 2 (x + y) 2 j e bx + y g 2 ex − y j 2 bx − yg = bx + yg bx + y g 2 + 2xy − y 2 − y 2 + 2xy − x 2 2 2 ∴ 3 ∂2 f ∂2 f = .

2 2 F− x I GH y2 JK F xI 1−G J H yK 1 2 ex + xy 2 + y2 FG 1IJ FG IJ H xK H K 1 y 1+ x 2 . (ii) j Adding (i) and (ii), we get x ∂u ∂u +y = 0, hence Euler’s theorem is verified. ∂x ∂y Example 6. If u = x sin–1 x2 F xI GH y JK + y sin–1 ∂ 2u ∂ 2u 2 + 2xy ∂x∂y ∂x Sol. We have + y2 ∂u = sin–1 ∂x ∂ 2u . t. x and y respectively. (v) x Multiplying equation (iv) by x and (v) by y and adding, we get x2 Example 7. If u = tan–1 x2 ∂ 2u ∂ 2u ∂ 2u 2 + 2xy + y = 0. ∂y2 ∂x∂y ∂x2 x 3 + y3 ∂u ∂u , prove that x +y = sin 2u and evaluate x− y ∂x ∂y ∂ 2u ∂ 2u + 2xy + y2 ∂x∂y ∂x2 ∂ 2u .

I) x + y2 . + 2 2 F− x I GH y2 JK F xI 1−G J H yK 1 2 ex + xy 2 + y2 FG 1IJ FG IJ H xK H K 1 y 1+ x 2 . (ii) j Adding (i) and (ii), we get x ∂u ∂u +y = 0, hence Euler’s theorem is verified. ∂x ∂y Example 6. If u = x sin–1 x2 F xI GH y JK + y sin–1 ∂ 2u ∂ 2u 2 + 2xy ∂x∂y ∂x Sol. We have + y2 ∂u = sin–1 ∂x ∂ 2u . t. x and y respectively. (v) x Multiplying equation (iv) by x and (v) by y and adding, we get x2 Example 7. If u = tan–1 x2 ∂ 2u ∂ 2u ∂ 2u 2 + 2xy + y = 0. ∂y2 ∂x∂y ∂x2 x 3 + y3 ∂u ∂u , prove that x +y = sin 2u and evaluate x− y ∂x ∂y ∂ 2u ∂ 2u + 2xy + y2 ∂x∂y ∂x2 ∂ 2u .

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

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