Solve differential equation ty’+(t+1)y=t , y(ln2)=1 , t>0

TyplusTplus1Y equal T

In this question, we have to find the Integration of the given function ty+(t+1)y=t by using different integration rules.

The basic concept behind this question is the knowledge of derivatives, integration, and the rules such as the product and quotient integration rules.

Expert Answer

Given function we have:

ty+(t+1)y=t

First, divide t into both sides of the equation and then we will get:

1t×ty+1t×(t+1)y=1t×t

Canceling t in the numerator with the denominator we get:

y+(t+1)ty=1

We know that here y=dydx, putting in the equation:

dydx+(t+1)ty=1

We also know that:

$p(t)=(t+1)t ; q(t)=1$

Putting these in our equation, we will have:

dydx+p(t)y=q(t)

Now let us suppose:

u(t)=ep(t)dt

After putting the value of p(t) here then we will have:

u(t)=e(t+1)tdt

Integrating the power of e:

u(t)=ettdt+1tdt

u(t)=et+ln(t)

Now we will simplify the exponential equation as follows:

u(t)=tet

From the second law of logarithm:

u(t)=elntet

Take log on both sides of the equation:

lnu(t)=lnelntet

lnu(t)=lntet

u(t)=tet

We know that:

y(x)=u(t)q(t)dtu(t)

y(x)=(tet)(1)dttet

y(x)=tetdttet

Using integration by parts:

tetdt=tetet+c

y(x)=tetet+ctet

y(x)=tettetettet+ctet

y(x)=11t+ctet

Putting the initial condition:

1=11ln2+cln2et

1ln2=cln2et

ln2etln2=c1

eln2=c

c=2

Substituting the value of c in the equation:

y(x)=11t+ctet

y(x)=11t+2tet

Numerical Result

y(x)=11t+2tet

Example

Integrate the following function:

1xdx

Solution:

=ln|x|

=elnx

We know that elnx=x so we have the above equation as:

=x

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