Solve the following differential equation for the given initial conditions. ydy/

Question

Solve the following differential equation for the given initial conditions. ydy/dt = t square root 1 + y^2 where y(0) = 0 Solve xdy/dx = x^2 + xy + y^2/x with y(1) = 1 Find the general solution for the 2^nd order linear DE: d^2y/dx^2 + y = csc(x) Solve the following nonhomogenous system of differential equations dy_1/dt - 1y_1 - 3y_2 = t dy_2/dt + 3y_1 - 1y_2 = 5 Solve the following differential equation using the method of power series. This is the only method that will be accepted. d^2y/dx^2 + (1 - x^2) dy/dx + 3y = 0 Determine the recursive relationship between the coefficients. Calculate at least the first 5 non-zero coefficients starting from a_o in terms of two unknown constants. Write out the solution y(x) with the coefficients you calculated.

Explanation / Answer

1) y dy/dt =t(1+y2)

y dy/(1+y2)  =tdt

integrate on both sides

y dy/(1+y2)  = tdt

let 1+y2=u ==>2y dy =du ==>y dy =(1/2)du

y dy/(1+y2) = (1/2)du/u= (1/2)2u +c =u +c =(1+y2)

y dy/(1+y2)  = tdt

==>(1+y2) =(1/2)t2+c

y(0)=0

==>(1+02) =(1/2)02+c

==>1=0+c

==>c=1

==>(1+y2) =(1/2)t2+1

==>(1+y2) =((1/2)t2+1)2

==>(1+y2) =((1/4)t4+t2+1)

==>y2 =(1/4)t4+t2

==>y =[(1/4)t4+t2]

==>y =(1/2)t (t2+4)

Related Questions

Navigate

• Browse (All)
• Browse S
• Subjects

---

---

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