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7.2 definite integral — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMaths7.2 definite integral1 Mark
MCQ
If $\int\limits_{a}^{x}{ty(t)dt={{x}^{2}}+y(x)}$ then $y$ as a function of $x$ is
  • ✓
    $y = 2 -(2 + a^2) {e^{\frac{{{x^2} - {a^2}}}{2}}}$
  • B
    $y = 1 -(2 + a^2) {e^{\frac{{{x^2} - {a^2}}}{2}}}$
  • C
    $y = 2 -(1 + a^2) {e^{\frac{{{x^2} - {a^2}}}{2}}}$
  • D
    none
✓

Answer

Correct option: A.
$y = 2 -(2 + a^2) {e^{\frac{{{x^2} - {a^2}}}{2}}}$
a
Given $\int_{a}^{x} t y(t) d t=x^{2}+y(x)$

$\Rightarrow x y=2 x+\frac{d y}{d x}$

$\Rightarrow x(y-2)=\frac{d y}{d x}$

$\Rightarrow \int x d x=\int \frac{d y}{y-2}$

$\Rightarrow \frac{x^{2}}{2}=\ln |y-2|+\ln c$

$\Rightarrow e^{\frac{x^{2}}{2}}=c(y-2) \quad-(2)$

Now, put $x=a$ in eqn ( 1$)$

$\Rightarrow y=-a^{2}$

So, by eqn ( 2$)$ $\therefore e^{\frac{a^{2}}{2}}=c\left(-a^{2}-2\right)$

$\Rightarrow c=-\frac{e^{\frac{a^{2}}{2}}}{\left(a^{2}+2\right)}$

Put this value in (2) $\therefore e^{\frac{x^{2}}{2}}=-\frac{e^{\frac{a^{2}}{2}}}{\left(a^{2}+2\right)}(y-2)$

$\Rightarrow-y+2=\left(a^{2}+2\right) e^{\frac{x^{2}-a^{2}}{2}}$

$\Rightarrow y=2-\left(2+a^{2}\right) e^{\frac{x^{2}-a^{2}}{2}}$

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