Solve the following differential equations: dr dt =-rt, r(0)=r_ 0 - Vidyadip

Question

Solve the following differential equations:$\frac{\text{dr}}{\text{dt}}=-\text{rt, r}(0)=\text{r}_{0}$

✓

Answer

$\frac{\text{dr}}{\text{dt}}=-\text{rt, r}(0)=\text{r}_{0}$
$\int\frac{\text{dr}}{\text{r}}=-\int\text{tdt}$
$\log|\text{r}|=-\frac{\text{t}^2}{2}+\text{C}...(1)$
Put $\text{t = 0, r = r}_{0}$ inequation (1),
$\log|\text{r}_0|=0+\text{C}$
$\log|\text{r}_0|=\text{C}$
Now,
$\log|\text{r}|=-\frac{\text{t}^2}{2}+\log|\text{r}_0|$
$\frac{\text{r}}{\text{r}_0}=\text{e}^{-\frac{\text{t}^2}{2}}$
$\text{r}=\text{r}_0\text{e}^{-\frac{\text{t}^2}{2}}$

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 "Solve the Following Question.(3 Marks)" from Differential Equations→Differential Equations — all question types→Maths STD 12 Science question papers→Maharashtra Board STD 12 Science question papers→Maharashtra Board question paper generator→

Similar questions

Find the distance between the parallel lines $\frac{x}{2}=\frac{y}{-1}=\frac{z}{2}$ and $\frac{x-1}{2}=\frac{y-1}{-1}=\frac{z-1}{2}$.

Find the vector equation of the plane passing through the point $A(-1,2,-5)$ and parallel to the vectors $4 \hat{i}-\hat{j}+3 \hat{k}$ and $\hat{i}+\hat{\jmath}-\hat{k}$.

Find the combined equation of the pair of lines passing through the origin and perpendicular to the lines represented by $3 x^2+2 x y-y^2=0$.

Evaluate the following integrals:
$\int\frac{\tan\text{x}}{\sec\text{x}+\tan\text{x}}\text{dx}$

Evalute the following integrals:
$\int\frac{\sec\text{x}}{\sec2\text{x}}\text{dx}$

Find the acute angle between lines $\frac{x-1}{1}=\frac{y-2}{-1}=\frac{z-3}{2}$ and $\frac{x-1}{2}=\frac{y-2}{1}=\frac{z-3}{1}$.

Evaluate the following integrals:$\int\text{e}^{\text{}x}\Big(\frac{1+\sin\text{x}}{1+\cos\text{x}}\Big)\text{dx}$

Find $\Big[\vec{\text{a}}\ \vec{\text{b}}\ \vec{\text{c}}\Big]$, when
$\vec{\text{a}}=\hat{\text{i}}-2\hat{\text{j}}+3\hat{\text{k}},\vec{\text{b}}=2\hat{\text{i}}+\hat{\text{j}}-\hat{\text{k}}$ and $\vec{\text{c}}=\hat{\text{j}}+\hat{\text{k}}$

Find the slope of the normal at the point 't' on the curve $\text{x}=\frac{1}{\text{t}},\text{y}=\text{t}.$

Integrate the following functions w. r. t. x

$\int \frac{1}{2 \sin 2 x-3} \cdot d x$