Question 8,Solve the initial value and determine the interval of existence 1.4 S

Question

Question 8,Solve the initial value and determine the interval of existence

1.4 Separable dy-3t2 + 4m2, dt2(y-l) y(0)=-1 dy-tsiny 10, cos y dt = 1+12, y(1)=m/2 11· dt=k(a-y)(b-y), y(0)=0, a,b>0 12.31 dt*)cost, y(1)-0 13. Any differential equation of the form dy/dt -f(y) is separable. T solve all those first-order differential equations in which time does explicitly. Now, suppose we have a differential equation of the fo f(y/1), such as, for example, the equation dy/dl=sin(y/1). Differ tions of this form are called homogeneous equations. Since the righ only depends on the single variable y/, t suggests itself to maket tion y/t=u or y = to. (a) Show that this substitution replaces the equation dy/dt-fO equivalent equation tdo/dt o-f(v), which is separable. (b) Find the general solution of the equation dy/dt=2(y/1)+(y/ as a function of the single variable y/t. (a) +21y 14. Determine whether each of the following functions of t and y can b (b) 2.3

Explanation / Answer

8.(1+t2)(1/2)(dy/dt)=ty3(1+t2)(-1/2)

dy/dt=(ty3(1+t2)(-1)

dy/y3=(t/(1+t2))dt

integrating both the sides

-(1/2)y-2=1/2 log(1 + t2) +const

given y(0)=1

so for t= 0, y= 1

putting this in the above equation

-(1/2)=const

-(1/2)y-2=1/2 log(1 + t2) +(-1/2)

y-2=1- log(1 + t2) Ans.

Related Questions

Navigate

• Browse (All)
• Browse Q
• Subjects

---

---

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