If (a+bx)e^y/x=x then prove that x(d²y)/(dx²)= ((a)/(a+bx) )².

x(d²y)/(dx²)= ((a)/(a+bx) )² (proved)

MathsCBSE Class 12 · Sample paper, session 2023-24 · 3 marks · Short answer

CBSE Class 12 Mathematics sample paper 2023-24: question 31

From the official CBSE sample question paper for 2023-24, section C, question 31, for 3 marks. Try it first; the hints and our step-by-step solution are below.

Q31 · 3 marks · Short answerDifferential Equations

If $(a + b x) e^{y / x} = x$ then prove that $x \frac{d ^{2} y}{d x ^{2}} = (\frac{a}{a + b x})^{2}$ .

Source: official sample paper from cbseacademic.nic.in (PDF). See it in the full Sample paper 2023-24.

Hints

Open one hint at a time and try again before the next.

Hint 1

Take logarithms first so that $y / x$ stands alone.

Hint 2

Simplify to $x \frac{d y}{d x} - y = \frac{a x}{a + b x}$ before differentiating again.

Step-by-step solution

Each step shows the marks it earns, as an examiner would award them. Checked by an independent re-solve from the question paper.

$e^{y / x} = \frac{x}{a + b x} \Rightarrow \frac{y}{x} = lo g x - lo g (a + b x)$ .
½ mark
Differentiate with respect to $x$ : $\frac{x \frac{d y}{d x} - y}{x ^{2}} = \frac{1}{x} - \frac{b}{a + b x}$ .
1 mark
$\Rightarrow x \frac{d y}{d x} - y = x^{2} \cdot \frac{a + b x - b x}{x ( a + b x )} = \frac{a x}{a + b x}$ .
½ mark
Differentiate again: $x \frac{d ^{2} y}{d x ^{2}} + \frac{d y}{d x} - \frac{d y}{d x} = \frac{( a + b x ) a - a x \cdot b}{( a + b x ) ^{2}}$ .
½ mark
$\Rightarrow x \frac{d ^{2} y}{d x ^{2}} = \frac{a ^{2}}{( a + b x ) ^{2}} = (\frac{a}{a + b x})^{2}$ . Hence proved.
½ mark

Answer: $x \frac{d ^{2} y}{d x ^{2}} = (\frac{a}{a + b x})^{2}$ (proved)

Where the marks are

Takes logs to get $\frac{y}{x} = lo g x - lo g (a + b x)$ (½ mark)
Differentiates correctly once (1 mark)
Simplifies to $x y^{'} - y = \frac{a x}{a + b x}$ (½ mark)
Differentiates a second time (½ mark)
Simplifies to the required result (½ mark)

Revise the chapter

Chapter: Differential Equations (notes, formulas, where marks are lost and every past question)

Practise this chapterBoard-style questions from Differential Equations, one at a time, marked against the answer key and marking points.Back to the full paperSample paper 2023-24: every question, by section.

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CBSE Class 12 Mathematics sample paper 2023-24: question 31

Hints

Step-by-step solution

Revise the chapter

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