Rate 5 stars for correct answer and steps. Sometimes a change of variable can be

Question

Rate 5 stars for correct answer and steps.

Sometimes a change of variable can be used to convert a differential equation y' = f(t, y) into a separable equation. One common change of variable technique is as follows. Consider a differential equation of the form y' = f (alpha t + beta y + gamma ), where alpha , beta , and gamma are constants. Use the change of variable Z = alpha t + beta y + gamma to rewrite the differential equation as a separable equation of the form z' = g(z). Solve the initial value problem g(z) = help (formulas) y(t) = help (formulas)

Explanation / Answer

z = t+y -> z' = 1+y'

z'-1 = (y+t)^2-1 = z^2-1

z' = z^2

g(z) = z^2

b)

z' = z^2

1/(z^2) dz = dt

int 1/z^2 dz = int dt

-1/z = t + C

z = -1/(t+C)

y+t = -1/(t+C)

y = -1/(t+C) - t

y(1) = -1/(C+1) - 1 = 6 -> C+1 = -1/7 -> C = -8/7

So we have:

y(t) = -1/(t - 8/7) - t

Related Questions

Navigate

• Browse (All)
• Browse R
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.