Differential Equations
Linear DE with x as Function of y
nta_pyq_2024_jan
Grade 12
Question:
If $\dfrac{dx}{dy}=\dfrac{1+x-y^2}{y}$, $x(1)=1$, then $5x(2)$ is equal to:
Step-by-Step Solution
Key Concept: Rewrite as $\frac{dx}{dy}-\frac{x}{y}=\frac{1-y^2}{y}$. I.F. $=e^{-\int\frac{1}{y}dy}=\frac{1}{y}$. Integrate, apply $x(1)=1$ to find $c$, then evaluate $5x(2)$.
I.F.$=\frac{1}{y}$: $\frac{x}{y}=\frac{-1}{y}-y+c$. $x=-1-y^2+cy$. $x(1)=1\Rightarrow c=3$. $x(2)=-1-4+6=1$. $5x(2)=5$.
Correct Answer: 5