HW1solutions - Differential Equations Assignment 1:...

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Unformatted text preview: Differential Equations Assignment 1: Explicit Equations and Qualitative study of first order equations. 1. Consider the following explicit differential equations, solve these equations and draw a sketch of some of the possible solutions: (a) dy dx = x 2 Integrating both sides with respect to x we obtain: Z dy dx dx = Z x 2 dx y = x 3 3 + C Figure 1: Slope field and Phase Portrait 1a 1 (b) dy dx = 1 Integrating both sides with respect to x we obtain: Z dy dx dx = Z 1 dx y = x + C Figure 2: Slope field and Phase Portrait 1b (c) dy dx = sin( x ) Integrating both sides with respect to x we obtain: Z dy dx dx = Z sin( x ) dx y =- cos( x ) + C 2 Figure 3: Slope field and Phase Portrait 1c 2. Consider the following autonomous differential equations, draw a stability diagram and slope field for each equation: (Stability diagrams have been drawn without the stability arrows) (a) dy dx = y + 2 (b) dy dx = 3- y (c) dy dx = y 2- 1 (d) dy dx = sin( y ) 3 Figure 4: Stability diagram and Slope field 2a Figure 5: Stability diagram and Slope field 2b 4 Figure 6: Stability diagram and Slope field 2c Figure 7: Stability diagram and Slope field 2d 5 3. Below is a list of several different differential equations classify these equations as to3....
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HW1solutions - Differential Equations Assignment 1:...

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