Solve the linear differential equation $\displaystyle\frac{dy}{dx} + y = x$.

Solve the linear differential equation (dy)/(dx) + y = x.

Exercises · Q15

\displaystyle\frac{dy}{dx} + y = x

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Standard linear form with $P = 1$ and $Q = x$ .

Integrating factor: $\int P\,dx = \int 1\,dx = x$ , so $\text{I.F.} = e^{x}$ .

General solution:

y\cdot e^{x} = \int x\,e^{x}\,dx + c.

Integrate $\int x e^{x}\,dx$ by parts (with $u=x$ , $dv=e^{x}dx$ ): $x e^{x} - \int e^{x}\,dx = x e^{x} - e^{x} = e^{x}(x-1)$ . Hence

y\,e^{x} = e^{x}(x - 1) + c.

Divide by $e^{x}$ :

$y = x - 1 + c\,e^{-x}.$ …

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Same / Similar Concept — real previous-year questions on the same or a closely similar concept, not this exact question.

CBSE 2025Set ANNUAL1 markMCQ
$\frac{dy}{dx} + \frac{y}{x} = x^3 - 3$
(a) $\log x$
(b) $e^x$
(c) $\frac{1}{x}$
(d) $x$
›Reveal solutionSolution

Identify $P = \frac{1}{x}$ in the linear form $\frac{dy}{dx}+Py=Q$ , then $\text{I.F.} = e^{\int P\,dx} = e^{\log x} = x$ .

The given equation

$\frac{dy}{dx} + \frac{y}{x} = x^3 - 3$

is a linear differential equation of the standard form $\dfrac{dy}{dx} + Py = Q$ , with

$P = \frac{1}{x}, \qquad Q = x^3 - 3.$

…

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CBSE 2023Set ANNUAL1 markMCQ
$\frac{dy}{dx} + y = e^{-x}$
(a) $x$
(b) $-x$
(c) $e^x$
(d) $e^{-x}$
›Reveal solutionSolution

For $\dfrac{dy}{dx} + y = e^{-x}$ we have $P = 1$ , so $\text{I.F.} = e^{\int 1\,dx} = e^{x}$ .

A first-order linear differential equation has the standard form:

$\frac{dy}{dx} + Py = Q,$

where $P$ and $Q$ are functions of $x$ (or constants). Its integrating factor (I.F.) is $e^{\int P\,dx}$ .

…

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CBSE 2022Set ANNUAL1 markMCQ
$\dfrac{dy}{dx} + \dfrac{y}{x} = x^3$
(a) True
(b) False
›Reveal solutionSolution

The equation is linear with $P(x) = \dfrac{1}{x}$ , so its integrating factor is $e^{\int \frac{1}{x}\,dx} = e^{\log x} = x$ , not $-x$ . The statement is False.

The given equation $\dfrac{dy}{dx} + \dfrac{y}{x} = x^3$ is a linear differential equation of the form $\dfrac{dy}{dx} + P(x)\,y = Q(x)$ with $P(x) = \dfrac{1}{x}$ .

The integrating factor (I.F.) is …

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Q.Solve the linear differential equation $\displaystyle\frac{dy}{dx} + y = x$ .

Concept understanding — Linear Differential Equations

Q.Solve the linear differential equation dydx+y=x\displaystyle\frac{dy}{dx} + y = xdxdy​+y=x.

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Q.Solve the linear differential equation $\displaystyle\frac{dy}{dx} + y = x$ .

Unlock everything free for 14 days