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Question: Find the differential equation by eliminating constants \(a,b\) from \(xy = a{e^x} + b{e^{ - x}}\) i...

Find the differential equation by eliminating constants a,ba,b from xy=aex+be−xxy = a{e^x} + b{e^{ - x}} is
(a) xy2+2y1+xy=0\left( a \right){\text{ x}}{{\text{y}}_2} + 2{y_1} + xy = 0
(b) xy2−2y1+xy=0\left( b \right){\text{ x}}{{\text{y}}_2} - 2{y_1} + xy = 0
(c) xy2+2y1−xy=0\left( c \right){\text{ x}}{{\text{y}}_2} + 2{y_1} - xy = 0
(d) xy2−2y1−xy=0\left( d \right){\text{ x}}{{\text{y}}_2} - 2{y_1} - xy = 0

Explanation

Solution

Here we have to find the differential equations and for this, we will differentiate the equation two times with respect to xx. So by doing so we can easily eliminate the constants a,ba,b from xy=aex+be−xxy = 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=aex+be−xxy = a{e^x} + b{e^{ - x}} and we will name it equation 11
Now on differentiating the above equation from both the side with respect to xx, we get
⇒y+xy1=aex−be−x\Rightarrow y + x{y_1} = a{e^x} - b{e^{ - x}}, let’s name it equation 22
Now on adding both the equations, we get
⇒xy+y+xy1=2aex\Rightarrow xy + y + x{y_1} = 2a{e^x}, we will name it equation 33
Now on differentiating the above equation again both the sides with respect to xx, we get
⇒y+xy1+y1+y1+xy2=2aex\Rightarrow y + x{y_1} + {y_1} + {y_1} + x{y_2} = 2a{e^x}, and we will name it equation 44
Now on comparing the equation 33 and equation 44, we get
⇒xy+y+xy1=y+xy1+y1+y1+xy2\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
⇒−xy+2y1+xy2=0\Rightarrow - xy + 2{y_1} + x{y_2} = 0
And on arranging the terms, we can also write it as
⇒xy2−xy+2y1=0\Rightarrow x{y_2} - xy + 2{y_1} = 0
There we will see that the order and degree of the differential equation will be 11.
Hence, we had successfully eliminated the constant terms a,ba,b from xy=aex+be−xxy = a{e^x} + b{e^{ - x}}.
Therefore, the option (c)\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 ex{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.