Question

The curve represented by the differential equation \[xdy - ydx = ydy\], intersects the y-axis at A (0,1) and the line \[y = e\] at (a, b,), then the value of \[\left( {a + b} \right)\] is

Answer 1

Hint: A differential equation is an equation which contains one or more terms and the derivatives of one variable (i.e., dependent variable) with respect to the other variable (i.e., independent variable), as to find out the value of \[\left( {a + b} \right)\], differentiate the given equation with respect to y and apply logarithmic functions to solve the function.

Complete step by step solution:
The given differential equation is \[xdy - ydx = ydy\].
Divide both sides of the equation by \[{y^2}\] as
\[\dfrac{{xdy - ydx}}{{{y^2}}} = \dfrac{{ydy}}{{{y^2}}}\]
\[\Rightarrow\dfrac{{xdy - ydx}}{{{y^2}}} = \dfrac{{dy}}{y}\]
As the differential equation intersects the y-axis
\[\dfrac{{ydx - xdy}}{{{y^2}}} = - \dfrac{{dy}}{y}\]
\[\Rightarrow d\left( {\dfrac{x}{y}} \right) = \dfrac{{ - dy}}{y}\]
Differentiating the terms, we get
\[\dfrac{x}{y} = - \log y + C\]
\[\Rightarrow\dfrac{x}{y} = \log \left( {\dfrac{1}{y}} \right) + C\]
$\Rightarrow 0 = 0 + C$
Implies that $C=0$.
Now substitute y=e in
\[\dfrac{x}{y} = \log \left( {\dfrac{1}{y}} \right)\]
\[\Rightarrow\dfrac{x}{e} = \log \left( {\dfrac{1}{e}} \right)\]
After differentiating we get the value of x as
\[\dfrac{x}{e} = - 1\]
\[\Rightarrow x = - e\], \[\left( { - e,e} \right)\]
\[\therefore\left( {a + b} \right)= 0 \]

Hence, \[\left( {a + b} \right)\] is zero.

Additional information:
Order of Differential Equation: The order of the highest order derivative present in the differential equation is called the order of the equation. If the order of the differential equation is 1, then it is called first order. If the order of the equation is 2, then it is called a second-order, and so on.

Degree of Differential Equation: The degree of the differential equation is the power of the highest order derivative, where the original equation is represented in the form of a polynomial equation in derivatives such as \[y'\], \[y''\],\[y'''\] and so on.

Note: A differential equation contains derivatives which are either partial derivatives or ordinary derivatives. The derivative represents a rate of change, and the differential equation describes a relationship between the quantity that is continuously varying with respect to the change in another quantity.