Question

Find the differential equation by eliminating constants $a,b$ from $xy = a{e^x} + b{e^{ - x}}$ is
$\left( a \right){\text{ x}}{{\text{y}}_2} + 2{y_1} + xy = 0$
$\left( b \right){\text{ x}}{{\text{y}}_2} - 2{y_1} + xy = 0$
$\left( c \right){\text{ x}}{{\text{y}}_2} + 2{y_1} - xy = 0$
$\left( d \right){\text{ x}}{{\text{y}}_2} - 2{y_1} - xy = 0$

Answer 1

Hint: Here we have to find the differential equations and for this, we will differentiate the equation two times with respect to \[x\]. So by doing so we can easily eliminate the constants $a,b$ from $xy = a{e^x} + b{e^{ - x}}$. After the first differentiation, we will add both the equations, and then we will differentiate it again and then solve it for the elimination.

Complete step-by-step answer:
 So we have the equation given us $xy = a{e^x} + b{e^{ - x}}$ and we will name it equation $1$
Now on differentiating the above equation from both the side with respect to \[x\], we get
$ \Rightarrow y + x{y_1} = a{e^x} - b{e^{ - x}}$, let’s name it equation $2$
Now on adding both the equations, we get
$ \Rightarrow xy + y + x{y_1} = 2a{e^x}$, we will name it equation $3$
Now on differentiating the above equation again both the sides with respect to \[x\], we get
$ \Rightarrow y + x{y_1} + {y_1} + {y_1} + x{y_2} = 2a{e^x}$, and we will name it equation $4$
Now on comparing the equation $3$ and equation $4$, we get
$ \Rightarrow xy + y + x{y_1} = y + x{y_1} + {y_1} + {y_1} + x{y_2}$
Now on solving the above equation and canceling the common terms, we get
$ \Rightarrow - xy + 2{y_1} + x{y_2} = 0$
And on arranging the terms, we can also write it as
$ \Rightarrow x{y_2} - xy + 2{y_1} = 0$
There we will see that the order and degree of the differential equation will be $1$.
Hence, we had successfully eliminated the constant terms $a,b$ from $xy = a{e^x} + b{e^{ - x}}$.
Therefore, the option $\left( c \right)$ is correct.

Note: We can also solve it by differentiating the equation two times initially and then putting the value we can get the eliminated equation. Remember one thing whenever we are dealing with an equation in terms ${e^x}$ , and we have to eliminate the constants in the equation, always go on differentiating the equation, we will always get the way to eliminate the constants.