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Question

$y equals fraction numerator x over denominator x plus 1 end fraction$ is a solution of the differential equation —.

$y^{2} \frac{d y}{d x} = x^{2}$
$x^{2} \frac{d y}{d x} = y^{2}$
$y \frac{d y}{d x} = x$
$x \frac{d y}{d x} = y$

hint Hint:

We are given a solution of differential equation. We have to find the differential equation of which it's solution. We will different the solution to get the differential equation.

The correct answer is: $x to the power of 2 end exponent fraction numerator d y over denominator d x end fraction equals y to the power of 2 end exponent$

The given solution is $y space equals fraction numerator x over denominator 1 space plus space x end fraction$
We will differentiate the given equation using u by v method.
$fraction numerator d y over denominator d x end fraction equals fraction numerator left parenthesis 1 space plus space x right parenthesis begin display style fraction numerator d x over denominator d x end fraction end style minus x begin display style fraction numerator d over denominator d x end fraction end style left parenthesis x space plus space 1 right parenthesis over denominator left parenthesis x space plus space 1 right parenthesis squared end fraction space space space space space space space equals space fraction numerator left parenthesis 1 space plus space x right parenthesis 1 space minus space x left parenthesis 1 right parenthesis over denominator left parenthesis x space plus space 1 right parenthesis squared end fraction space space space space space space space equals fraction numerator 1 space plus space x space minus space x over denominator left parenthesis x space plus space 1 right parenthesis squared end fraction space space space space space space space equals 1 over left parenthesis x space plus space 1 right parenthesis squared$
Now, we will substitute the value for (x + 1)²
$y space equals fraction numerator x over denominator left parenthesis x space plus space 1 right parenthesis end fraction M u l t i p l y i n g space b o t h space t h e space s i d e s space b y space left parenthesis x space plus space 1 right parenthesis space a n d space d i v i d i n g space b y space y. left parenthesis x space plus space 1 right parenthesis space equals x over y$
We will substitute this value in the above equation.
$fraction numerator d y over denominator d x end fraction equals 1 over open parentheses begin display style x over y end style close parentheses squared fraction numerator d y over denominator d x end fraction equals y squared over x squared M u l t i p l y i n g space b o t h space t h e space s i d e s space b y space x squared x squared fraction numerator d y over denominator d x end fraction equals y squared$
This is the required differential equation.

For such questions, we should know different method to solve differential equation.

Related Questions to study

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