Download App

DIFFERENTIAL EQUATIONS — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDIFFERENTIAL EQUATIONS3 Marks
Question
Find the general solution of $\frac{\text{dy}}{\text{dx}}+\text{ay}=\text{e}^{\text{mx}}.$
✓

Answer

We have, $\frac{\text{dy}}{\text{dx}}+\text{ay}=\text{e}^{\text{mx}}$
which is a linear differential equation.
On comparing it with $\frac{\text{dy}}{\text{dx}}+\text{Py}=\text{Q},$ we get
$\text{P}=\text{a},\text{Q}=\text{e}^{\text{mx}}$
$\text{I.F.}=\text{e}^{\int\text{P}\text{dx}}$
$\text{I.F.}=\text{e}^{\int\text{a}\text{dx}}$
$\text{I.F.}=\text{e}^{\text{ax}}$
The general solution is,
$\text{y.}\text{e}^{\text{ax}}=\int\text{e}^{\text{mx}}.\text{e}^{\text{ax}}\text{dx}+\text{C}$
$\Rightarrow\text{y.}\text{e}^{\text{ax}}=\int\text{e}^{(\text{m}+\text{a})\text{x}}\text{ dx}+\text{C}$
$\Rightarrow\text{y}.\text{e}^\text{ax}=\frac{\text{e}^{(\text{m}+\text{a})\text{x}}}{(\text{m}+\text{a})}+\text{C}$

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" 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

Find fog and gof if:

$\text{f}(\text{x})=\text{c},\text{c}\in \text{R},\text{g(x)}=\sin \text{x}^2$

→
If $\cos y = x\cos \left( {a + y} \right)$ with $\cos a \ne \pm 1$ prove that $\frac{{dy}}{{dx}} = \frac{{{{\cos }^2}\left( {a + y} \right)}}{{\sin a}}$ 
→
Evaluate the following integrals:
$\int\frac{\sec^2\sqrt{\text{x}}}{\sqrt{\text{x}}}\text{ dx}$
→
Evaluate the following integrals:

$\int\frac{\text{e}^\text{x}}{(1+\text{e}^{\text{x}})(2+\text{e}^\text{x})}\text{dx}$

→
Evaluate the following integrals:
$\int^\limits{\frac{\pi}{4}}_{\frac{-\pi}{4}}|\sin\text{x}|\text{dx}$
→
The line $\vec{\text{r}}=\hat{\text{i}}+\lambda(2\hat{\text{i}}-\text{m}\hat{\text{j}}-3\hat{\text{k}})$ is parallel to the plane $\vec{\text{r}}\cdot(\text{m}\hat{\text{i}}+3\hat{\text{j}}+\hat{\text{k}})=4.$ Find m.
→
Evaluate the following integrals:
$\int^\limits{\frac{\pi}{2}}_{\frac{-\pi}{2}}\big\{\sin|\text{x}|+\cos|\text{x}|\big\}\text{dx}$
→
Show that the following planes are at right angles.
x - 2y + 4z = 10 and 18x + 17y + 4z = 49
→
Ten eggs are drawn successively, with replacement, from a lot containing 10% defective eggs. Find the probability that there is at least one defective egg.
→
Find the second-order derivative of the function sin(log x)
→