How reduce the steps of lengthy answer (partial fraction decomposition)

[PA3040D]

Dear expert, I have answered the exam question attached below, but I believe that the answer is correct. However, the process is lengthy and time-consuming. Could you please assist me in simplifying the steps that reducing the time required?

Find the inverse Laplace of

 

[mario99]

As my experience tells me, there is no shortcut for doing partial fraction decomposition by hand. But with time, you will be quicker.

 

[Dr.Peterson]

Dear expert, I have answered the exam question attached below, but I believe that the answer is correct. However, the process is lengthy and time-consuming. Could you please assist me in simplifying the steps that reducing the time required?

There is, in fact, a shortcut for finding partial fractions when the denominators are distinct linear factors:

(By the way, technically you omitted the denominator on the first line, but the ultimate equation is correct.)

What you do is to set [imath]s[/imath] to each of the zeros of the denominators, one at a time:

[imath]s=1[/imath]: [imath](1)^2-6(1)+10=A(1-2){\color{Red}(1-1)}+B(1-3){\color{Red}(1-1)}+C(1-3)(1-2)\implies 5={\color{Red}0}+{\color{Red}0}+2C\implies C=5/2[/imath]

[imath]s=2[/imath]: [imath](2)^2-6(2)+10=A{\color{Red}(2-2)}(2-1)+B(2-3)(2-1)+C(2-3){\color{Red}(2-2)}\implies 2={\color{Red}0}-B+{\color{Red}0}\implies B=-2[/imath]

[imath]s=3[/imath]: [imath](3)^2-6(3)+10=A(3-2)(3-1)+B{\color{Red}(3-3)}(3-1)+C{\color{Red}(3-3)}(3-2)\implies 1=2A+{\color{Red}0}+{\color{Red}0}\implies A=1/2[/imath]

 

[PA3040D]

There is, in fact, a shortcut for finding partial fractions when the denominators are distinct linear factors:
View attachment 38013
(By the way, technically you omitted the denominator on the first line, but the ultimate equation is correct.)

What you do is to set [imath]s[/imath] to each of the zeros of the denominators, one at a time:

[imath]s=1[/imath]: [imath](1)^2-6(1)+10=A(1-2){\color{Red}(1-1)}+B(1-3){\color{Red}(1-1)}+C(1-3)(1-2)\implies 5={\color{Red}0}+{\color{Red}0}+2C\implies C=5/2[/imath]

[imath]s=2[/imath]: [imath](2)^2-6(2)+10=A{\color{Red}(2-2)}(2-1)+B(2-3)(2-1)+C(2-3){\color{Red}(2-2)}\implies 2={\color{Red}0}-B+{\color{Red}0}\implies B=-2[/imath]

[imath]s=3[/imath]: [imath](3)^2-6(3)+10=A(3-2)(3-1)+B{\color{Red}(3-3)}(3-1)+C{\color{Red}(3-3)}(3-2)\implies 1=2A+{\color{Red}0}+{\color{Red}0}\implies A=1/2[/imath]

This is fine ...Grate thanks

 

[Dr.Peterson]

This is fine ...Grate thanks

A quick matter of English: The word you probably mean is "great", not "grate". Isn't English fun?

But, you're welcome.