✓ Board-verified
Mathematics · 2026 · 3 marks
Differential EquationsMethods of Solving First Order, First Degree Differential Equations3 marksApplyshort answer
CBSE 2026 · Region 1 · Set 1 · Q29
(a) Find the general solution of the following differential equation: x^2 dy/dx=x^2+x y + y^2
(a)
Find the general solution of the following differential equation : \[x^{2} \frac{\mathrm{dy}}{\mathrm{dx}}=x^{2}+x \mathrm{y}+\mathrm{y}^{2}
\]
OR
(b)
Find the particular solution of the differential equation
$\displaystyle x \mathrm{y} \frac{\mathrm{dy}}{\mathrm{d} x}=(x+2)(\mathrm{y}+2)$, given that $\displaystyle \mathrm{y}(1)=-1$.
Official answer
(a)
y = x·tan(ln|x| + C) or equivalent form; (b) ln|y+$\displaystyle 2$| - ln|x| = 2ln|y+$\displaystyle 1$| + ln($\displaystyle 3$) or equivalent implicit form
Marking-scheme solution
Given differential equation is :$\displaystyle \dfrac{dy}{dx} = 1+\dfrac{\mathrm{y}}{x}+\left(\dfrac{\mathrm{y}}{x}\right)^2$
Put $\displaystyle \mathrm{y}=vx \Rightarrow \dfrac{dy}{dx} = v+x\dfrac{dv}{dx}$
The given differential equation becomes $\displaystyle x\dfrac{dv}{dx} = 1+v^2$
$\displaystyle \Rightarrow \dfrac{dv}{1+v^2} = \dfrac{dx}{x}$
$\displaystyle \Rightarrow \int\dfrac{dv}{1+v^2} = \int\dfrac{dx}{x}$
$\displaystyle \Rightarrow \tan^{-1}v = \log|x|+C$
General solution: $\displaystyle \tan^{-1}\left(\dfrac{\mathrm{y}}{x}\right) = \log|x|+C$
Given differential equation is: $\displaystyle \mathrm{xy} \frac{\mathrm{dy}}{\mathrm{d} x}=(x+2)(\mathrm{y}+2)$
$\displaystyle \Rightarrow \dfrac{\mathrm{y}}{\mathrm{y}+2}\,dy = \dfrac{x+2}{x}\,dx$
$\displaystyle \Rightarrow \int\left(1-\dfrac{2}{\mathrm{y}+2}\right)dy = \int\left(1+\dfrac{2}{x}\right)dx$
$\displaystyle \Rightarrow \mathrm{y}-2\log|\mathrm{y}+2| = x+2\log|x|+C$
At $\displaystyle x=1,\mathrm{y}=-1$: $\displaystyle C=-2$
Particular solution: $\displaystyle \mathrm{y}-2\log|\mathrm{y}+2| = x+2\log|x|-2$, or $\displaystyle \mathrm{y} = x+2\log|x(\mathrm{y}+2)|-2$
All Differential Equations questions2026 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 2026 board exam, with the answer as CBSE’s own marking scheme gives it. Where our answers come from.