For each of the differential equations in find the general soluti… - Vidyadip

Question

For each of the differential equations in find the general solution:
$(e^x + e^{–x}) ~dy – (e^x – e^{–x}) dx = 0$

✓

Answer

Given: Differential equation $(e^x + e^{–x}) ~dy – (e^x – e^{–x}) dx = 0$
$\Rightarrow \ (\text{e}^\text{x} + \text{e}^{-\text{x}}) \ \text{dy} = (\text{e}^\text{x} – \text{e}^{–\text{x}}) \ \text{dx} \ $ $\Rightarrow \ \text{dy}=\Bigg(\frac{\text{e}^\text{x}-\text{e}^{-\text{x}}}{{\text{e}^\text{x}+\text{e}^{-\text{x}}}}\Bigg) \text{dx}$
Integrating both sides, $\ \int\text{dy}=\int\Bigg(\frac{\text{e}^\text{x}-\text{e}^{-\text{x}}}{\text{e}^\text{x}+\text{e}^{-\text{x}}}\Bigg)\text{dx}$
$\Rightarrow\ \text{y}=\text{log}\Big|\text{e}^\text{x}+\text{e}^{-\text{x}}\Big|+\text{c} \ \Bigg[\because\int\frac{f'(\text{x})}{f(\text{x})}\text{dx}=\text{log}\big|f(\text{x})\big|\Bigg]$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "3 Marks Question" from DIFFERENTIAL EQUATIONS→DIFFERENTIAL EQUATIONS — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Write the value of $\cos^{-1}\Big(\tan\frac{3\pi}{4}\Big).$

Prove that $y = \frac{{4\sin \theta }}{{\left( {2 + \cos \theta } \right)}} - \theta $ is an increasing function of $\theta$ in $\left[ {0,\frac{\pi }{2}} \right]$

Let I be any interval disjoint from [-1,1 ]. Prove that the function f given by f(x) = x + $\frac{1}{\text{x}}$ is strictly increasing on I.

Evaluate the following:
$\tan^{-1}1+\cos^{-1}\Big(-\frac{1}{2}\Big)+\sin^{-1}\Big(-\frac{1}{2}\Big)$

Find the area of the triangle with vertices at the points:
$(0, 0), (6, 0)$ and $(4, 3)$

Evaluate the following integrals:
$\int^\limits{\pi}_{0}5\big(5-4\cos\theta\big)^{\frac{1}{4}}\sin\theta\text{ d}\theta$

Show that the points whose position vectors are as given below are collinear:
$3\hat{\text{i}}-2\hat{\text{j}}+4\hat{\text{k}},\ \hat{\text{i}}+\hat{\text{j}}+\hat{\text{k}}$ and $-\hat{\text{i}}+4\hat{\text{j}}-2\hat{\text{k}}$

Eight coins are thrown simultaneously. Find the chance of obtaining at least six heads.

If $l_1, m_1, n_1$ and $l_2, m_2, n_2$ are the direction cosines of two mutually perpendicular lines, show that the direction cosines of the line perpendicular to both of these are $m_1n_2 - m_2n_1, l_1n_2 - l_2n_1, l_1m_2 - l_2m_1$.

Evaluate the following integrals:
$\int (3\text{x}\sqrt{\text{x}}+4\sqrt{\text{x}}+5)\text{dx}$