Name: Formation of Differential Equations
Uploaded: 2019-07-19
Duration: 11 min 9 s
Description: In this video, we are going to learn about the formation of differential equations in detail. Master this concept in just a few minutes with this quick lesson.

Question

Show that $y = e^{x} + e^{2 x}$ is the solution of the initial value problem $\frac{d^{2} y}{d x^{2}} - 3 \frac{d y}{d x} + 2 y = 0, y (0) = 2, y^{'} (0) = 3$ .

Accepted Answer

The given function is $y = e^{x} + e^{2 x} . . . . . . . . . . (i)$

Differentiating it with respect to $x$ ,

$\frac{d y}{d x} = e^{x} + 2 e^{2 x} . . . . . . . . . . . (ii)$

Again, differentiating it with respect to $x$ ,

$\Rightarrow \frac{d^{2} y}{d x^{2}} = e^{x} + 4 e^{2 x}$

We have to prove that $\frac{d^{2} y}{d x^{2}} - 3 \frac{d y}{d x} + 2 y = 0$

Consider $L . H . S . = \frac{d^{2} y}{d x^{2}} - 3 \frac{d y}{d x} + 2 y$

$= e^{x} + 4 e^{2 x} - 3 (e^{x} + 2 e^{2 x}) + 2 (e^{x} + e^{2 x})$

$= e^{x} + 4 e^{2 x} - 3 e^{x} - 6 e^{2 x} + 2 e^{x} + 2 e^{2 x}$

$= 0$

$= R . H . S .$

Hence, $\frac{d^{2} y}{d x^{2}} - 3 \frac{d y}{d x} + 2 y = 0$

Thus, $y = e^{x} + 2 e^{2 x}$ satisfies the differential equation $\frac{d^{2} y}{d x^{2}} - 3 \frac{d y}{d x} + 2 y = 0$ .

Put $x = 0$ in the equation $(i)$ ,

$y (0) = e^{0} + e^{0}$

$\Rightarrow y (0) = 1 + 1 = 2$

Put $x = 0$ in the equation $(ii)$ ,

$y^{'} (0) = e^{0} + 2 e^{0}$

$\Rightarrow y^{'} (0) = 1 + 2 = 3$

Hence, $y = e^{x} + 2 e^{2 x}$ is the solution of the given initial value problem.

Show that $y = e^{x} + e^{2 x}$ is the solution of the initial value problem $\frac{d^{2} y}{d x^{2}} - 3 \frac{d y}{d x} + 2 y = 0, y (0) = 2, y^{'} (0) = 3$ .

Important Questions on Differential Equations

Learn and Prepare for any exam you want !

Show that y=ex+e2x is the solution of the initial value problem d2ydx2-3dydx+2y=0, y(0)=2, y'(0)=3.

Important Questions on Differential Equations

Learn and Prepare for any exam you want !

Show that $y = e^{x} + e^{2 x}$ is the solution of the initial value problem $\frac{d^{2} y}{d x^{2}} - 3 \frac{d y}{d x} + 2 y = 0, y (0) = 2, y^{'} (0) = 3$ .