CBSE 2023 · Region 2 · Set 1 · Q29 · 3 marks
Find the general solution of the differential equation : \[\left(\mathrm{x} \mathrm{y}-\mathrm{x}^{2}\right) d \mathrm{y}=\mathrm{y}^{2} d \mathrm{x} \]Find the general solution of the differential equation : \[\left(\mathrm{x}^{2}+1\right) \frac{\mathrm{dy}}{\mathrm{dx}}+2 \mathrm{x} \mathrm{y}=\sqrt{\mathrm{x}^{2}+4} \]
Find the general solution of the differential equation : \[\left(\mathrm{x} \mathrm{y}-\mathrm{x}^{2}\right) d \mathrm{y}=\mathrm{y}^{2} d \mathrm{x} \]
Find the general solution of the differential equation : \[\left(\mathrm{x}^{2}+1\right) \frac{\mathrm{dy}}{\mathrm{dx}}+2 \mathrm{x} \mathrm{y}=\sqrt{\mathrm{x}^{2}+4} \]
Marking-scheme solution
(a)
Given differential equation can be written as
$$\frac{d \mathrm{y}}{d \mathrm{x}}=\frac{\mathrm{y}^{$\displaystyle 2$}}{\mathrm{x} \mathrm{y}-\mathrm{x}^{$\displaystyle 2$}}=\frac{\left(\dfrac{\mathrm{y}}{\mathrm{x}}\right)^{$\displaystyle 2$}}{\dfrac{\mathrm{y}}{\mathrm{x}}-$\displaystyle 1$}
$$Separating the variables and integrating
$$\int\left($\displaystyle 1$-\frac{1}{u}\right) d u=\int \frac{d \mathrm{x}}{\mathrm{x}} \Rightarrow u-\log u=\log |\mathrm{x}|+c \Rightarrow \frac{\mathrm{y}}{\mathrm{x}}-\log \frac{\mathrm{y}}{\mathrm{x}}=\log |\mathrm{x}|+c
$$Or
Rewriting the given differential equation as:
$$\frac{d \mathrm{y}}{d \mathrm{x}}+\frac{$\displaystyle 2$ \mathrm{x}}{$\displaystyle 1$+\mathrm{x}^{$\displaystyle 2$}} \mathrm{y}=\frac{\sqrt{\mathrm{x}^{$\displaystyle 2$}+$\displaystyle 4$}}{$\displaystyle 1$+\mathrm{x}^{$\displaystyle 2$}}
$$\text { Integrating factor }=\mathrm{e}^{\int \frac{$\displaystyle 2$ \mathrm{x}}{$\displaystyle 1$+\mathrm{x}^{$\displaystyle 2$}} d \mathrm{x}}=\mathrm{e}^{\log \left($\displaystyle 1$+\mathrm{x}^{$\displaystyle 2$}\right)}=$\displaystyle 1$+\mathrm{x}^{$\displaystyle 2$}
$$∴ solution of the differential equation is
Differential EquationsMethods of Solving First Order, First Degree Differential EquationsApplyshort_answermedium
More from Differential Equations
- The integrating factor of the differential equation (1-y^2) (d x)/(dy)+y x=ay,(-1<y<1) is2023 · asked 3×
- The integrating factor of the differential equation (x+2 y^2) (d y)/(d x)=y(y 0) is:2024 · asked 3×
- The integrating factor of differential equation R (d x)/(dy)+P x=Q where P, Q, R are functions of y is2026 · asked 3×
- Camphor is a waxy, colourless solid with strong aroma that evaporates through the process of sublimation, if…2025 · asked 3×
- A bacteria sample of certain number of bacteria is observed to grow exponentially in a given amount of time.…2024 · asked 3×
- Degree of the differential equation sin x+ cos ((d y)/(d x))=y^2 is2023 · asked 3×
- If p and q are respectively the order and degree of the differential equation /(d x)((dy)/(d x))^3=0, then…2025 · asked 3×
- Find the particular solution of the differential equation (dy)/(dx)=(x+y)/(x), y(1)=0. OR Find the general…2023 · asked 3×
CBSE Class 12 Mathematics past-paper question from the 2023board exam, with the answer as CBSE’s own marking scheme gives it. Where our answers come from.