For each of the differential equations given in find the general …

Question

For each of the differential equations given in find the general solution:
$(\text{x}+\text{y})\frac{\text{dy}}{\text{dx}}=1$

✓

Answer

$(\text{x}+\text{y})\frac{\text{dy}}{\text{dx}}=1$
$\Rightarrow\frac{\text{dy}}{\text{dx}}=\frac{1}{\text{x}+\text{y}}$
$\Rightarrow\frac{\text{dx}}{\text{dy}}=\text{x}+\text{y}$
$\Rightarrow\frac{\text{dx}}{\text{dy}}-\text{x}=\text{y}$
This is a linear differential equation of the form:
$\frac{\text{dy}}{\text{dx}}+\text{px}=\text{Q}\ (\text{where p}=-1\ \text{and}\ \text{Q}=\text{y})$
$\text{Now, I.F}=\text{e}^{\int\text{pdy}}=\text{e}^{\int-\text{dy}}=\text{e}^{-\text{y}}.$
The general solution of the given differential equation is given by the relation,
$\text{x(I.F)}=\int(\text{Q}\times\text{I.F.})\text{dy}+\text{C}$
$\Rightarrow\text{xe}^{-\text{y}}=\int(\text{y}\cdot\text{e}^{-\text{y}})\text{dy}+\text{C}$
$\Rightarrow\text{xe}^{-\text{y}}=\text{y}\cdot\int\text{e}^{-\text{y}}\text{dy}-\int\Big[\frac{\text{d}}{\text{dy}}(\text{y})\int\text{e}^{-\text{y}}\text{dy}\Big]\text{dy}+\text{C}$
$\Rightarrow\text{xe}^{-\text{y}}=\text{y}(-\text{e}^{-\text{y}})-\int(-\text{e}^{-\text{y}})\text{dy}+\text{C}$
$\Rightarrow\text{xe}^{-\text{y}}=-\text{ye}^{-\text{y}}+\int\text{e}^{-\text{y}}\text{dy}+\text{C}$
$\Rightarrow\text{xe}^{-\text{y}}=-\text{ye}^{-\text{y}}-\text{e}^{-\text{y}}+\text{C}$
$\Rightarrow\text{x}=-\text{y}-1+\text{Ce}^{\text{y}}$
$\Rightarrow\text{x}+\text{y}+1=\text{Ce}^{\text{y}}$

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 "4 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

Evaluate the following integrals:
$\int(\tan^{-1}\text{x}^2)\text{x dx}$

→

Finde the value of a and b, if the function f(x) defined by $\text{f(x)}\begin{cases}\text{x}^2+3\text{x}+\text{a}, &\text{x}\leq1\\\text{bx}+2, & \text{x}>1\end{cases}$is differentiable at x = 1.

→

Find the area, lying above x-axis and included between the circle x² + y² = 8x and the parabola y² = 4x.

→

Write the values of 'a' for which the following distribution of probabilities becomes a probability distributioin:

$\text{X}=\text{x}_\text{i}:$	$-2$	$-1$	$0$	$1$
$\text{P}(\text{X}=\text{x}_\text{i}):$	$\frac{1-\text{a}}{4}$	$\frac{1+2\text{a}}{4}$	$\frac{1-2\text{a}}{4}$	$\frac{1+\text{a}}{4}$