By Smirnov V.I.

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**Additional resources for A course of higher mathematics, vol. 1**

**Sample text**

11. (a) If u1 and u2 are solutions, so is u = c1 u1 + c2 u2 since u′′ − 4 u′ + 3 u = c1 (u′′ 1 − ′ x 3x 4 u′1 + 3 u1 ) + c2 (u′′ − 4 u + 3 u ) = 0. (b) span { e , e }; (c) 2. 12. Each is a solution, and the general solution u(x) = c1 + c2 cos x + c3 sin x is a linear combination of the three independent solutions. 13. (a) e2 x ; (b) cos 2 x, sin 2 x; (c) e3 x , 1; (d) e− x , e− 3 x ; (e) e− x/2 cos 23 x, √ √ √ √ √ x x x x e− x/2 sin 23 x; (f ) e5 x , 1, x; (g) ex/ 2 cos √ , ex/ 2 sin √ , e− x/ 2 cos √ , e− x/ 2 sin √ .

0 a11 B B a21 det B @ a31 a41 1 a12 a13 a14 a22 a23 a24 C C C= a32 a33 a34 A a42 a43 a44 a11 a22 a33 a44 − a11 a22 a34 a43 − a11 a23 a32 a44 + a11 a23 a34 a42 − a11 a24 a33 a42 + a11 a24 a32 a43 − a12 a21 a33 a44 + a12 a21 a34 a43 + a12 a23 a31 a44 − a12 a23 a34 a41 + a12 a24 a33 a41 − a12 a24 a31 a43 + a13 a21 a32 a44 − a13 a21 a34 a42 − a13 a22 a31 a44 + a13 a22 a34 a41 − a13 a24 a32 a41 + a13 a24 a31 a42 − a14 a21 a32 a43 + a14 a21 a33 a42 + a14 a22 a31 a43 − a14 a22 a33 a41 + a14 a23 a32 a41 − a14 a23 a31 a42 .

Moreover, every v = + = can y 0 y be written as a sum of vectors on the two axes. = 3 y is x = y! = 0, the lines only ! 3b a x , where a = − 21 x+ 32 y, + = intersect at the origin. Moreover, every v = b a y b = 21 x − 12 y, can be written as a sum of vectors on each line. (c) A vector v = ( a, 2 a, 3 a )T in the line belongs to the plane if and only if a + 2 (2 a) + 3 (3 a)0 =114 a = 0 0, so a = 0 and the only 0 common element is1v = 0. Moreover, every 1 x + 2 y + 3 z x 1 B 13 x − 2 y − 3 z C 1 B C C v=B @ 2 (x + 2 y + 3 z) A + @ − 2 x + 10 y − 6 z A can be written as a sum @yA = 14 3 (x + 2 y + 3 z) 14 −3x − 6y + 5z z of a vector in the line and a vector in the plane.

### A course of higher mathematics, vol. 1 by Smirnov V.I.

by Michael

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