Question

- Find the P.I. of the differential equation(D2-7D+10)y=2x +x2
(A) ex
(B) 1/2.e
(C) both A and B true
(D) none of these​

Answer 1

Answer:

none of these is write ans

Answer 2

The given differential equation is

$\quad (D^{2}-7D+10)y=2x+x^{2}$

To find C.F.

Let the auxiliary equation be

$\quad m^{2}-7m+10=0$

$\Rightarrow (m-5)(m-2)=0$

$\Rightarrow m=5,2$

Thus C.F. is $C_{1}e^{5x}+C_{2}e^{2x}$.

To find P.I.

Let, $y_{p}=Ax^{2}+Bx+C$

Then $Dy_{p}=2Ax+B$

and $D^{2}y_{p}=2A$

Since $y_{p}$ is also a solution of the given differential equation,

$\quad D^{2}y_{p}-7Dy_{p}+10y_{p}=2x+x^{2}$

$\Rightarrow 2A-14Ax-7B+10Ax^{2}+10Bx+10C=2x+x^{2}$

Comparing among coefficients, we get

$\quad 2A-7B+10C=0$

$\quad -14A+10B=2$

$\quad 10A=1$

Solving, we get

• $A=\frac{1}{10},B=\frac{17}{50},C=\frac{109}{500}$

Thus, $y_{p}=\frac{1}{10}x^{2}+\frac{17}{50}x+\frac{109}{500}$

CHOOSING FROM OPTIONS:

Therefore, (D) none of these is the correct answer.

However,

the complete solution is

$y=C_{1}e^{5x}+C_{2}e^{2x}+\frac{1}{10}x^{2}+\frac{17}{50}x+\frac{109}{500}$