Question

If $\int\limits_{ a }^{ x } ty ( t ) dt = x ^2+ y ( x )$ then $y$ as a function of $x$ is

• A. $y=2-\left(2+a^2\right) e^{\frac{x^2-a^2}{2}}$
• B. $y=1-\left(2+a^2\right) e^{\frac{x^2-a^2}{2}}$
• C. $y=2-\left(1+a^2\right) e^{\frac{x^2-a^2}{2}}$
• D. none

Answer 1

Answer: (A)

$\int\limits_a^x y(t) d t=x^2+y(x)$
$\Rightarrow x y=2 x+\frac{d y}{d x} \Rightarrow x(y-2)=\frac{d y}{d x}$
$\Rightarrow \int x d x=\int \frac{d y}{y-2} \Rightarrow \frac{x^2}{2}=\ell n|y-2|+\ell n c$
$\Rightarrow e^{\frac{x^2}{2}}=c(y-2)$
$a t x=a y=-a^2$
$\therefore e^{\frac{a^2}{2}}=c\left(-a^2-2\right) \Rightarrow c=-\frac{e^{\frac{a^2}{2}}}{\left(a^2+2\right)}$
$\therefore e^{\frac{x^2}{2}}=-\frac{e^{\frac{a^2}{2}}}{\left(a^2+2\right)}(y-2)$
$\Rightarrow-y+2=\left(a^2+2\right) e^{\frac{x^2-a^2}{2}}$
$\Rightarrow y=2-\left(2+a^2\right) e^{\frac{x^2-a^2}{2}}$