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The Wronskian of two differential functions f and g is W(f.g)=fg'-gf, more formally, f(x) W(f,g)(x) = det [F(2) g(2)] = f(x)g' (2) — 9(2)ƒ(z). More generally, the Wronsikian of three functions f,g, and h is defined as f(x) g(x) h(x) W(f, g, h)(x) = det f'(x) g'(x) h'(x) [f"(x) g"(x) h" (x)] For example, W(sin(x),cos(x))= This shows that the Wronskian can be constant. Further, we can use properties of to compute the Wronskian. For example, W(cos(x),sin(x))= W(2 sin(x)+2022 cos(x),cos(x))= and W(cos(x),3 sin(x)+5 cos(x))=

The Wronskian of two differential functions f and g is W(f.g)=fg'-gf, more formally, f(x) W(f,g)(x) = det [F(2) g(2)] = f(x)g' (2) — 9(2)ƒ(z). More generally, the Wronsikian of three functions f,g, and h is defined as f(x) g(x) h(x) W(f, g, h)(x) = det f'(x) g'(x) h'(x) [f"(x) g"(x) h" (x)] For example, W(sin(x),cos(x))= This shows that the Wronskian can be constant. Further, we can use properties of to compute the Wronskian. For example, W(cos(x),sin(x))= W(2 sin(x)+2022 cos(x),cos(x))= and W(cos(x),3 sin(x)+5 cos(x))=

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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The Wronskian of two differential functions f and g is W(f.g)=fg'-gf, more formally, f(x) W(f,g)(x) = det [F(2) g(2)] = f(x)g' (2) — 9(2)ƒ(z). More generally, the Wronsikian of three functions f,g, and h is defined as f(x) g(x) h(x) W(f, g, h)(x) = det f'(x) g'(x) h'(x) [f"(x) g"(x) h" (x)] For example, W(sin(x),cos(x))= This shows that the Wronskian can be constant. Further, we can use properties of to compute the Wronskian. For example, W(cos(x),sin(x))= W(2 sin(x)+2022 cos(x),cos(x))= and W(cos(x),3 sin(x)+5 cos(x))=
Transcribed Image Text:The Wronskian of two differential functions f and g is W(f.g)=fg'-gf, more formally, f(x) W(f,g)(x) = det [F(2) g(2)] = f(x)g' (2) — 9(2)ƒ(z). More generally, the Wronsikian of three functions f,g, and h is defined as f(x) g(x) h(x) W(f, g, h)(x) = det f'(x) g'(x) h'(x) [f"(x) g"(x) h" (x)] For example, W(sin(x),cos(x))= This shows that the Wronskian can be constant. Further, we can use properties of to compute the Wronskian. For example, W(cos(x),sin(x))= W(2 sin(x)+2022 cos(x),cos(x))= and W(cos(x),3 sin(x)+5 cos(x))=
The case of three functions is more interesting. We have W(sin(x),cos(x),x)= and W(cos(x), sin(x),x)= Similarly, W(2 sin(x)+2022 cos(x),cos(x),x)= and W(cos(x),3 sin(x)+5 cos(x),x)= In some cases, given the Wronsikian we determine values of parameters appearing in functions. For example, 13z W(e¹¹, e³¹, x²) = 6e¹³ª (1 − 13x +20x²) 2 where A= and B= -3 -2x 3 2x 0 -3x 5 determinants -X X 8 3x -2 2 -1
Transcribed Image Text:The case of three functions is more interesting. We have W(sin(x),cos(x),x)= and W(cos(x), sin(x),x)= Similarly, W(2 sin(x)+2022 cos(x),cos(x),x)= and W(cos(x),3 sin(x)+5 cos(x),x)= In some cases, given the Wronsikian we determine values of parameters appearing in functions. For example, 13z W(e¹¹, e³¹, x²) = 6e¹³ª (1 − 13x +20x²) 2 where A= and B= -3 -2x 3 2x 0 -3x 5 determinants -X X 8 3x -2 2 -1
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