A curve passes through the origin and the slope of the tangent to the curve at any point $\left(x_{1}\right)$ is equal to the sum of the coordinates of the point. Find the equation of the curve.

Home » A curve passes through the origin and the slope of the tangent to the curve at any point is equal to the sum of the coordinates of the point. Find the equation of the curve.

CBSE | Class 12 | Linear Differential Equations | Maths | RS Aggarwal

A curve passes through the origin and the slope of the tangent to the curve at any point $\left(x_{1}\right)$ is equal to the sum of the coordinates of the point. Find the equation of the curve.

Solution:

General solution for the differential equation in the form of $\frac{d y}{d x}+\mathrm{Py}=$ Qis given by,
$y \cdot(I . F .)=\int Q \cdot(I . F .) d x+c$
Where, integrating factor,
$\text { I.F. }=e \int^{p d x}$
The slope of the tangent to the curve $=\mathrm{dy} / \mathrm{dx}$
The slope of the tangent to the curve is equal to the sum of the coordinates of the point.
$\frac{d y}{d x}=x+y($ differential equation $)$
$\frac{d y}{d x}-y=x \ldots \ldots \ldots(1)$
In equation (1)
$\mathrm{P}=-1$ and $\mathrm{Q}=\mathrm{x}$
Therefore, integrating factor is
$\text { I.F. }=e \int^{p} d x=\mathrm{e}^{j-1 \mathrm{dx}}=\mathrm{e}^{-\mathrm{x}}$
General solution is
$\begin{array}{r} y \cdot(I . F \cdot)=\int Q \cdot(I . F \cdot) d x+c \\ \text { y. }\left(e^{-x}\right)=\int(x) \cdot\left(e^{-x}\right) d x+c \ldots \ldots .(2) \end{array}$
$I=\int(x) \cdot\left(e^{-x}\right) d x$
Let, $u=x$ and $v=e^{-x}$
$\begin{array}{l} \mathrm{I}=\mathrm{x} . \int e^{-x} d x-\int\left(\frac{d}{d x}(x) \cdot \int e^{-x} d x\right) d x \\ \mathrm{I}=-x \cdot e^{-x}-e^{-x} \end{array}$
Substituting I in (2),
$y \cdot\left(e^{-x}\right)=-x \cdot e^{-x}-e^{-x}+c$
Dividing above equation by $\mathrm{e}^{-\mathrm{x}}$ ,
$\mathrm{y}=-x-1+c . e^{x} \text { (general solution) }$
The curve passes through origin, therefore the above equation satisfies for $x=0$ and $y=0$
$\mathrm{c}=1$
Substituting c in general solution,
$y+x+1=e^{x}$