Let y=y_1(x) and y=y_2(x) be two distinct solutions of the differential equation [(d y)/(d…

Question

Let $y=y_{1}(x)$ and $y=y_{2}(x)$ be two distinct solutions of the differential equation $$\frac{d y}{d x}=x+y$$, with $y_{1}(0)=0$ and $y_{2}(0)=1$ respectively. Then, the number of points of intersection of $y=y_{1}(x)$ and $y=y_{2}(x)$ is

Accepted Answer

$${{dy} \over {dx}} = x + y$$

Let $x + y = t$

$$1 + {{dy} \over {dx}} = {{dt} \over {dx}}$$

$${{dt} \over {dx}} - 1 = t \Rightarrow \int {{{dt} \over {t + 1}} = \int {dx} } $$

$$\ln |t + 1| = x + C'$$

$|t + 1| = C{e^x}$

$|x + y + 1| = C{e^x}$

For $${y_1}(x),\,{y_1}(0) = 0 \Rightarrow C = 1$$

For $${y_2}(x),\,{y_2}(0) = 1 \Rightarrow C = 2$$

${y_1}(x)$ is given by $|x + y + 1| = {e^x}$

${y_2}(x)$ is given by $|x + y + 1| = 2{e^x}$

At point of intersection

${e^x} = 2{e^x}$

No solution

So, there is no point of intersection of ${y_1}(x)$ and ${y_2}(x)$.

Solutions · 1