Is the Solution to the Given IVP Unique?

[saripalli89]

Homework Statement

Solve the IVP and determine if the solution is unique and explain why.

Homework Equations

t*y'+(t-2)y=t^4*e^t, y(0)=0

The Attempt at a Solution

t*y'+(t-2)y=t^4*e^t

y'+ ((t-2)/t)y=t^3*e^t

integration factor=e^(integral((t-2)/t dt)=e^t/(t^2)

d/dy((e^t/t^2)*y=t*e^(2t)

integrate both sides

(e^t/t^2)*y=e^(2t)*(t/2-1/4)+C

y=((t^3)/2-(t^2)/4)*e^t+(t^2*C)/e^t

y(0)=0----> 0=(0/2-0/4)*e^0+(0*C)/e^0, so how would i find the value for C and can someone explain the concept of uniqueness? I couldn't really follow my russian professor

 

[mighty2000]

Hello,

To find the value of C, you can use the initial condition given in the problem. Since y(0)=0, we can substitute t=0 and y=0 into the equation y=((t^3)/2-(t^2)/4)*e^t+(t^2*C)/e^t.

This gives us 0=((0^3)/2-(0^2)/4)*e^0+(0^2*C)/e^0. Simplifying, we get 0=0+0*C, which means that C can be any value. Therefore, the solution is not unique and there are infinite possible solutions.

The concept of uniqueness in this context means that there is only one possible solution to the given initial value problem. In this case, since there are infinite possible solutions, the solution is not unique.

Hope this helps!