Download App

Differential Equations — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDifferential Equations5 Marks
Question
Solve the following differential equation:
$\text{y e}^{\frac{\text{x}}{\text{y}}}\text{dx}=\big(\text{xe}^{\frac{\text{x}}{\text{y}}}+\text{y}\big)\text{dy}$
✓

Answer

We have,
$\text{y e}^{\frac{\text{x}}{\text{y}}}\text{dx}=\big(\text{xe}^{\frac{\text{x}}{\text{y}}}+\text{y}\big)\text{dy}$
$\Rightarrow\ \frac{\text{dy}}{\text{dx}}=\frac{\text{xe}^{\frac{\text{x}}{\text{y}}}+\text{y}}{\text{y e}^{\frac{\text{x}}{\text{y}}}}$
$\Rightarrow\ \frac{\text{dy}}{\text{dx}}=\frac{\frac{\text{x}}{\text{y}}\text{e}^{\frac{\text{x}}{\text{y}}}+1}{\text{e}^{\frac{\text{x}}{\text{y}}}}$
$\Rightarrow\ \frac{\text{dy}}{\text{dx}}=\frac{\text{x}}{\text{y}}+\text{e}^{\frac{-\text{x}}{\text{y}}}$
This is a homogeneous differential equation.
Putting y = vx and $\frac{\text{dy}}{\text{dx}}=\text{v + y}\frac{\text{dv}}{\text{dx}}$, we get
$\text{v + y}\frac{\text{dv}}{\text{dx}}=\text{v + e}^{-\text{v}}$
$\Rightarrow\ \text{y}\frac{\text{dv}}{\text{dy}}=\text{e}^{-\text{v}}$
$\Rightarrow\ \text{e}^{\text{v}}\text{dv}=\frac{1}{\text{y}}\text{dy}$
Integrating both sides, we get
$\int\text{e}^{\text{v}}\text{dv}=\int\frac{1}{\text{y}}\text{dy}$
$\Rightarrow\ \text{e}^{\text{v}}=\log|\text{y}|+\text{C}$
Putting $\text{v}=\frac{\text{y}}{\text{x}}$, we get
$\Rightarrow\ \text{e}^{\frac{\text{x}}{\text{y}}}=\log|\text{y}|+\text{C}$
Hence, $\text{e}^{\frac{\text{x}}{\text{y}}}=\log|\text{y}|+\text{C}$ 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→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Find the direction cosines of the line $\frac{\text{x}+2}{2}=\frac{2\text{y}-7}{6}=\frac{5-\text{z}}{6}.$ Also, find the vector equation of the line through the point A(-1, 2, 3) and parallel to the given line.
→
A class has 15 students whose ages are 14, 17, 15, 14, 21, 19, 20, 16, 18, 17, 20, 17, 16, 19 and 20 years respectively. One student is selected in such a manner that each has the same chance to being selected and the age X of the selected student is recorded. What is the probability distribution of the random variable X.
→
Solve the following differential equations:$\frac{\text{dy}}{\text{dx}}=2\text{e}^{2\text{x}}\text{y}^2,\text{y}(0)=-1$
→
If $\text{x}\sin(\text{a}+\text{y})+\sin\text{a}\cos(\text{a}+\text{y})=0,$ prove that $\frac{\text{dy}}{\text{dx}}=\frac{\sin^2(\text{a}+\text{y})}{\sin\text{a}}.$
→
Differentiate the following functions with respect to x:
$\tan^{-1}\Big(\frac{\sqrt{1+\text{a}^2\text{x}^2-1}}{\text{ax}}\Big),\text{x}\neq0$
→
Find the area bounded by the curve $\text{y}=\sqrt{\text{x}},$ x = 2y + 3 in the first quadrant and x-axis.
→
Verify Rolle's theorem for the following function on the indicated intervals$f(x) = (x^2- 1)(x - 2)$ on $[-1, 2]$
→
Using differentials, find the approximate values of the following:
$\Big(\frac{17}{81}\Big)^{\frac{1}{4}}$
→
Using differentials, find the approximate values of the following:
$(80)^{\frac{1}{4}}$
→
Prove that the lines through A(0, -1, -1) and B(4, 5, 1) intersects the line through C(3, 9, 4) and D(-4, 4, 4). Also, find their point of intersection.
→