Consider the differential equationdy/dt= t − y.(a) Sketch the slope field of the differential equationdy/dt= t − y in the range −1 ≤ t ≤ 3, −1 ≤ y ≤ 3. As an aid,observe that the isocline of slope c is the line t − y = c, so the segments have slope c at points on the line y = t − c.(b) Show that y = t − 1 + Ce−t is a solution for all C. Since limt→∞e−t = 0, these solutions approach the particularsolution y = t − 1 as t →∞. Explain how this behavior is reflected in your slope field.
Consider the differential equationdy/dt= t − y.(a) Sketch the slope field of the differential equationdy/dt= t − y in the range −1 ≤ t ≤ 3, −1 ≤ y ≤ 3. As an aid,observe that the isocline of slope c is the line t − y = c, so the segments have slope c at points on the line y = t − c.(b) Show that y = t − 1 + Ce−t is a solution for all C. Since limt→∞e−t = 0, these solutions approach the particularsolution y = t − 1 as t →∞. Explain how this behavior is reflected in your slope field.
Consider the differential equationdy/dt= t − y.(a) Sketch the slope field of the differential equationdy/dt= t − y in the range −1 ≤ t ≤ 3, −1 ≤ y ≤ 3. As an aid,observe that the isocline of slope c is the line t − y = c, so the segments have slope c at points on the line y = t − c.(b) Show that y = t − 1 + Ce−t is a solution for all C. Since limt→∞e−t = 0, these solutions approach the particularsolution y = t − 1 as t →∞. Explain how this behavior is reflected in your slope field.
Consider the differential equation dy/dt = t − y. (a) Sketch the slope field of the differential equation dy/dt= t − y in the range −1 ≤ t ≤ 3, −1 ≤ y ≤ 3. As an aid, observe that the isocline of slope c is the line t − y = c, so the segments have slope c at points on the line y = t − c. (b) Show that y = t − 1 + Ce −t is a solution for all C. Since lim t→∞e−t = 0, these solutions approach the particular solution y = t − 1 as t →∞. Explain how this behavior is reflected in your slope field.
With integration, one of the major concepts of calculus. Differentiation is the derivative or rate of change of a function with respect to the independent variable.
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