Download App

DIFFERENTIAL EQUATIONS — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSDIFFERENTIAL EQUATIONS5 Marks
Question
Solve the following differential equations:
$\text{x}\frac{\text{dy}}{\text{dx}}=\text{x + y}$
✓

Answer

We have,
$\text{x}\frac{\text{dy}}{\text{dx}}=\text{x + y}$
$\Rightarrow\ \frac{\text{dy}}{\text{dx}}=\frac{\text{x + y}}{\text{x}}$
This is a homogeneous differential equation.
Putting y = vx and $\frac{\text{dy}}{\text{dx}}=\text{v + x}\frac{\text{dv}}{\text{dx}}$, we get
$\text{v + x}\frac{\text{dv}}{\text{dx}}=\frac{\text{x + vx}}{\text{x}}$
$\Rightarrow\ \text{v + x}\frac{\text{dv}}{\text{dx}}=1+\text{v}$
$\Rightarrow\ \text{x}\frac{\text{dv}}{\text{dx}}=1+\text{v}-\text{v}$
$\Rightarrow\ \text{x}\frac{\text{dv}}{\text{dx}}=1$
$\Rightarrow\ \text{dv}=\frac{1}{\text{x}}\text{dx}$
Integrating both sides, we get
$\int\text{dv}=\int\frac{1}{\text{x}}\text{dx}$
$\Rightarrow\ \text{v}=\log|\text{x}|+\text{C}$
Putting $\text{v}=\frac{\text{y}}{\text{x}}$, we get
$\Rightarrow\ \frac{\text{y}}{\text{x}}=\log|\text{x}|+\text{C}$
$\Rightarrow\ \text{y}=\text{x}\log|\text{x}|+\text{Cx}$
Hence, $\text{y}=\text{x}\log|\text{x}|+\text{Cx}$ is the required solution.

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 "5 Marks Questions" from DIFFERENTIAL EQUATIONS→DIFFERENTIAL EQUATIONS — all question types→MATHS STD 12 Science question papers→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

Find the particular solution of the differential equation $(\tan^{–1} y – x) dy =(1 + y^2 ) dx,$ given that when $x = 0, y = 0.$
→
Find the maximum slope of the curve $y = −x^{3 }+ 3x^{2 }+ 2x − 27.$
→
Evaluate the following integrals:$\int(\text{x}+1)\text{e}^{\text{x}}\log(\text{xe}^{\text{x}})\text{dx}$
→
Find the distance of the point (- 1, - 5, - 10), from the point of intersection of the line
$\overrightarrow{\text{r}}=(2\hat{\text{i}}-\hat{\text{j}}+2\hat{\text{k})}+\lambda(3\hat{\text{i}}+4\hat{\text{j}}+2\hat{\text{k})}\text{ and the plane}\overrightarrow{\text{r}}\cdot(\hat{\text{i}}-\hat{\text{j}}+\hat{\text{k})}=5.$
→
Solve the following differential equation:$4\frac{\text{dy}}{\text{dx}}+8\text{y}=5\text{e}^{-3\text{x}}$
→
If $\text{x}=\text{a}\sin\text{t}\ \text{and}\ \text{y}=\text{a}(\cos\text{t}+\log\tan\frac{\text{t}}{2}),$ find $\frac{\text{d}^2\text{y}}{\text{dx}^2}$
→
Show that the right circular cylinder of given surface and maximum volume is such that its height is equal to the diameter of the base.
→
Verify Rolle's theorem of the following function on the indicated interval
$\text{f}(\text{x})=\frac{6\text{x}}{\pi}-4\sin^{2}\text{x}\text{ on }\Big[0,\frac{\pi}{6}\Big]$
→
Examine the consistency of the system of equation x + y + z = 1; 2x + 3y + 2z = 2; ax + ay + 2az = 4
→
If A and B are two events such that,
$\text{P(A)}=\frac{7}{13},\text{P(B)}=\frac{9}{13}$ and $\text{P}(\text{A}\cap\text{B})=\frac{4}{13},$ then find $\text{P}(\overline{\text{A}}|\text{B}).$
→