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3. For each of the following equations, state the order and whether it is nonlinear, linear inhomogeneous, or linear homogeneous; provide reasons. (a) ut uxx+1=0 (b) ut -Uxx + xu=0 (c) ut -Uxxt + uux = 0 (d) U₁t - Uxx + x² = 0 (e) iu - Uxx+u/x = 0 (f)_ux(1+u²)¯¹⁄² +uy(1 + u²)−¹/² = 0 (g) ux+e³uy = 0 (h) U₁+Uxxxx + √1+u = 0

3. For each of the following equations, state the order and whether it is nonlinear, linear inhomogeneous, or linear homogeneous; provide reasons. (a) ut uxx+1=0 (b) ut -Uxx + xu=0 (c) ut -Uxxt + uux = 0 (d) U₁t - Uxx + x² = 0 (e) iu - Uxx+u/x = 0 (f)_ux(1+u²)¯¹⁄² +uy(1 + u²)−¹/² = 0 (g) ux+e³uy = 0 (h) U₁+Uxxxx + √1+u = 0

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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[Partial Differential Equations] How do you solve number 3?

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2. Which of the following operators are linear? (a) Lu=ux + xUy (b) Lu=ux + uu y (c) Lu= ux+u² (d) (e) Lu= √ Lu=ux + Uy + 1 V1+x2 (cos y)u. +uyxy – [arctan(x/y)]u - 3. For each of the following equations, state the order and whether it is nonlinear, linear inhomogeneous, or linear homogeneous; provide reasons. (a) ut -Uxx + 1 = 0 Ut (b) ut -Uxx + xu = 0 (c) UtUxxt + uux = 0 (d) utt - Uxx + x² = 0 (e) iu uxx+u/x = 0 = 0 (f) ux(1+u²)¯¹⁄² +uy(1 +u²)¯¹⁄² ; (g) ux+e³uy = 0 (h) U₁+Uxxxx + √√1+u = 0 4. Show that the difference of two solutions of an inhomogeneous linear equation Lu = g with the same g is a solution of the homogeneous equation Lu= 0.
Transcribed Image Text:2. Which of the following operators are linear? (a) Lu=ux + xUy (b) Lu=ux + uu y (c) Lu= ux+u² (d) (e) Lu= √ Lu=ux + Uy + 1 V1+x2 (cos y)u. +uyxy – [arctan(x/y)]u - 3. For each of the following equations, state the order and whether it is nonlinear, linear inhomogeneous, or linear homogeneous; provide reasons. (a) ut -Uxx + 1 = 0 Ut (b) ut -Uxx + xu = 0 (c) UtUxxt + uux = 0 (d) utt - Uxx + x² = 0 (e) iu uxx+u/x = 0 = 0 (f) ux(1+u²)¯¹⁄² +uy(1 +u²)¯¹⁄² ; (g) ux+e³uy = 0 (h) U₁+Uxxxx + √√1+u = 0 4. Show that the difference of two solutions of an inhomogeneous linear equation Lu = g with the same g is a solution of the homogeneous equation Lu= 0.
Linearity means the following. Write the equation in the form Lu = 0, where L is an operator. That is, if v is any function, Lu is a new function. For instance, L = a/ax is the operator that takes v into its partial derivative Ux. In Example 2, the operator L is L=ə/Əx yə/Əy. (Lu a/ax +ya/ay. (Lu = ux + yuy.) The definition we want for linearity is + L(u + v) = Lu + Lv L(cu) = cLu (3) for any functions u, v and any constant c. Whenever (3) holds (for all choices of u, v, and c), L is called linear operator. The equation Lu = 0 (4) is called linear if L is a linear operator. Equation (4) is called a homogeneous linear equation. The equation Lu = 8, (5) where g 0 is a given function of the independent variables, is called an inhomogeneous linear equation. For instance, the equation (cos xy²)ux - y²uy = tan(x² + y²) is an inhomogeneous linear equation. (6)
Transcribed Image Text:Linearity means the following. Write the equation in the form Lu = 0, where L is an operator. That is, if v is any function, Lu is a new function. For instance, L = a/ax is the operator that takes v into its partial derivative Ux. In Example 2, the operator L is L=ə/Əx yə/Əy. (Lu a/ax +ya/ay. (Lu = ux + yuy.) The definition we want for linearity is + L(u + v) = Lu + Lv L(cu) = cLu (3) for any functions u, v and any constant c. Whenever (3) holds (for all choices of u, v, and c), L is called linear operator. The equation Lu = 0 (4) is called linear if L is a linear operator. Equation (4) is called a homogeneous linear equation. The equation Lu = 8, (5) where g 0 is a given function of the independent variables, is called an inhomogeneous linear equation. For instance, the equation (cos xy²)ux - y²uy = tan(x² + y²) is an inhomogeneous linear equation. (6)
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