Differential Equations
Homogeneous DE — Shifting Origin
nta_pyq_2024_jan
Grade 12
Question:
If the solution curve of the differential equation $\dfrac{dy}{dx}=\dfrac{x+y-2}{x-y}$ passing through the point $(2,1)$ is $\tan^{-1}\left(\dfrac{y-1}{x-1}\right)-\dfrac{1}{\beta}\log_e\left(\alpha+\left(\dfrac{y-1}{x-1}\right)^2\right)=\log_e|x-1|$, then $5\beta+\alpha$ is equal to
Step-by-Step Solution
Key Concept: Shift origin: $X=x-1,Y=y-1$ to get $\frac{dY}{dX}=\frac{X+Y}{X-Y}$. This is homogeneous; let $Y=vX$, separate and integrate.
Shift to $X=x-1,Y=y-1$: $\frac{dY}{dX}=\frac{X+Y}{X-Y}$. Let $Y=vX$: $\frac{1-v}{1+v^2}dv=\frac{dX}{X}$. Integrate: $\tan^{-1}v-\frac{1}{2}\ln(1+v^2)=\ln|X|+C$. So $\alpha=1,\beta=2$. $5\beta+\alpha=11$.
Correct Answer: 11