Solve the following differential equation: x dy – y dx = x^ 2 +y^… - Vidyadip

Question

Solve the following differential equation:
x dy – y dx = $\sqrt{\text{x}^{2}+\text{y}^{2}}\text{ dx}$ .

✓

Answer

Given equation can be written as $\frac{\text{dy}}{\text{dx}}=\frac{\text{y}}{\text{x}}+\sqrt{1+\Big(\frac{\text{y}}{\text{x}}\Big)^{2}}$
$\Rightarrow\text{v + x }\frac{\text{dv}}{\text{dx}}=\text{v}+\sqrt{1+\text{v}^{2}}$ where $\frac{\text{y}}{\text{x}}=\text{v}$
$\Rightarrow\int\frac{\text{dv}}{\sqrt{1 + \text{v}^{2}}}=\int\frac{\text{dx}}{\text{x}}$
$\Rightarrow\log|\text{v}+\sqrt{1+\text{v}^{2}}|=\log\text{cx}$
$\Rightarrow\text{v}+\sqrt{1+\text{v}^{2}}=\text{cx}\therefore\text{y}+\sqrt{\text{x}^{2}+\text{y}^{2}}=\text{cx}^{2}$.

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 "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

If the area enclosed by the parabolas $y^2 - 16ax$ and $x^2 = 16ay, a > 0$ is $\frac{1024}{3}$ square units, find the value of $a.$

Let $\text{A}=\begin{bmatrix}1&-1&0\\2&1&3\\1&2&1\end{bmatrix}$ and $\text{B}=\begin{bmatrix}1&2&3\\2&1&3\\0&1&1\end{bmatrix},$ Find $A^T, B^T$ and verify that.$(\text{A}\text{B})^\text{T}=\text{B}^\text{T}+\text{A}^\text{T}$

Solve the following differential equation: $\frac{\text{dy}}{\text{dx}}+\text{y}=\sin\text{x}$

Find the area bounded by the parabola $y^2 = 4x$ and the line $y = 2x - 4:$
By using horizontal strips.

Form the differential equation corresponding to $\text{y}=\text{e}^{\text{mx}}$ by eliminating m.

Differentiate the following functions with respect to x:
$\tan^{-1}\Big(\frac{\sqrt{1+\text{a}^2\text{x}^2-1}}{\text{ax}}\Big),\text{x}\neq0$

Differentiate $\tan^{-1}\bigg(\frac{\sqrt{1 - \text{x}^{2}}}{\text{x}}\bigg)$with respect to $\cos^{-1}(2 \text{x}\sqrt{1- \text{x}^{2}}),$ when $\text{x}\neq0.$

If $\vec{\text{a}},\vec{\text{b}}$ are the position vectors of A, B respectively, find the position vector of a point C in AB produced such that AC = 3AB and that a point D in BA produced such that BD = 2BA.

Find a unit vector perpendicular to the plane A B C, where the coordinates of A, B and C are A (3, -1, 2), B (1, -1, -3) and C (4, -3, 1).

Differentiate the functions given in Exercise:
$\text{x}^{\sin\text{x}}+(\sin\text{x})^{\cos\text{x}}$