"Let y = y(x) be the solution of the differential equation dy = e$\alpha$x + y dx; $\alpha$ $\in$ N. If y(loge2) = loge2 and y(0) = loge$\left( {{1 \over 2}} \right)$, then the value of $\alpha$ is equal to _____________."

Question

Accepted Answer

"$\int {{e^{ - y}}} dy = \int {{e^{\alpha x}}} dx$

$\Rightarrow {e^{ - y}} = {{{e^{\alpha x}}} \over \alpha } + c$ ..... (i)

Put (x, y) = (ln2, ln2)

${{ - 1} \over 2} = {{{2^\alpha }} \over \alpha } + C$ ..... (ii)

Put (x, y) $\equiv$ (0, $-$ln2) in (i)

$- 2 = {1 \over \alpha } + C$ ..... (iii)

(ii) $-$ (iii)

${{{2^\alpha } - 1} \over \alpha } = {3 \over 2}$

$\Rightarrow$ $\alpha$ = 2 (as $\alpha$ $\in$ N)"

Let y = y(x) be the solution of the differential equation dy = e^{$\alpha$x + y} dx; $\alpha$ $\in$ N. If y(log_e2) = log_e2 and y(0) = log_e$\left( {{1 \over 2}} \right)$, then the value of $\alpha$ is equal to _____________.

Solution

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Let y = y(x) be the solution of the differential equation dy = e$\alpha$x + y dx; $\alpha$ $\in$ N. If y(loge2) = loge2 and y(0) = loge$\left( {{1 \over 2}} \right)$, then the value of $\alpha$ is equal to _____________.

Solution

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More Differential Equations questions

Drill 25 more like these. Every day. Free.

Let y = y(x) be the solution of the differential equation dy = e^{$\alpha$x + y} dx; $\alpha$ $\in$ N. If y(log_e2) = log_e2 and y(0) = log_e$\left( {{1 \over 2}} \right)$, then the value of $\alpha$ is equal to _____________.