✓ Board-verified
Mathematics · 2022 · 3 marks
Differential EquationsMethods of Solving First Order, First Degree Differential Equations3 marksApplyshort answer
CBSE 2022 · Region 3 · Set 1 · Q8
Find the particular solution of the differential equation x dy/(d x)- y =x^2 · e^x, given y (1)=0. OR Find the general solution of the differential…
Find the particular solution of the differential equation $\displaystyle x \frac{\mathrm{dy}}{\mathrm{d} x}-\mathrm{y}=x^{2} \cdot \mathrm{e}^{x}$, given $\displaystyle \mathrm{y}(1)=0$.
OR Find the general solution of the differential equation $\displaystyle x \frac{\mathrm{dy}}{\mathrm{d} x}=\mathrm{y}(\log \mathrm{y}-\log x+1)$.
Marking-scheme solution
\[\frac{dy}{dx} - \frac{y}{x} = \left(\frac{x^{2}}{x}\right)e^{x} = xe^{x}\]
\[I\cdot F = e^{\int -\frac{1}{x}dx} = e^{-\log x} = e^{\log x^{-1}} = \frac{1}{x}\]
Solution is $\displaystyle \displaystyle y \times \frac{1}{x} = \int e^{x}\,dx$
\[\frac{y}{x} = e^{x} + C\]
\[y = xe^{x} + Cx\]
\[C = -e\]Particular solution is
\[y = xe^{x} - ex\]
\[x\frac{dy}{dx} = y[\log y - \log x + 1]\]
\[\frac{dy}{dx} = \frac{y}{x}\left[\log \frac{y}{x} + 1\right]\]
\[y = vx\]
\[\frac{dy}{dx} = v + x\frac{dv}{dx}\]
\[v + x\frac{dv}{dx} = v[\log v + 1]\]
\[x\frac{dv}{dx} = v\log v\]
\[\int \frac{dv}{v\log v} = \int \frac{1}{x}\,dx\]
\[\log|\log v| = \log|x| + \log C\]
\[\log\left(\frac{y}{x}\right) = Cx\]
All Differential Equations questions2022 question paperNCERT solutions: Differential EquationsPractice Differential Equations →
More from Differential Equations
- The integrating factor of the differential equation (1-y^2) (d x)/dy+y x=ay,(-1<y<1) is2023
- The integrating factor for solving the differential equation x dy/dx-y=2 x^2 is:2023
- Product of the order and degree of differential equation 1+(dy/dx)^3=λ((d^3 y)/(dx^3))^2 is:2026
- Solve the differential equation (d y)/(d x)= cos x-2 y.2025
- The solution of the differential equation (d x)/x+(d y)/y=0 is:2023
- Find the general solution of the differential equation (y^2-x^2) d x=2 x y dy OR Find the particular solution of the differential equation (1+e^2 x)…2026
- The integrating factor of the differential equation (x+2 y^2) (d y)/(d x)=y(y>0) is:2024
- Find the particular solution of the differential equation dy/dx-y/x+cosec(y/x)=0; given that y=0, when x=1.2025
CBSE Class 12 Mathematics past-paper question from the 2022 board exam, with the answer as CBSE’s own marking scheme gives it. Where our answers come from.