Differential equation d^2y dx^2 - dy dx =0,y(0)=2,y'(0)=1 Functio… - Vidyadip

Question

Differential equation $\frac{\text{d}^2\text{y}}{\text{dx}^2}-\frac{\text{dy}}{\text{dx}}=0,\text{y}(0)=2,\text{y}'(0)=1$

Function $\text{y}=\text{e}^\text{x}+1$

✓

Answer

Here, $\text{y}=\text{e}^{\text{x}}+1 ....(1)$
Differentiating it with respect to x
$\frac{\text{dy}}{\text{dx}}=\text{e}^{\text{x}}$
$\frac{\text{dy}}{\text{dx}}=\text{y}-1 ...(2)$
Again, differentiating it with respect to x,
$\frac{\text{d}^{2}\text{y}}{\text{dx}^{2}}=\frac{\text{dy}}{\text{dx}}$
$\frac{\text{d}^{2}\text{y}}{\text{dx}^{2}}-\frac{\text{dy}}{\text{dx}}=0$
It is given differential equation. So,
y = e^x+ 1 is a solution of the equation
put x - 0 in equation (1)
⇒ y = e⁰+ 1 = 2
y(0) = 2
put x = 0 in equation (2)
y' = e⁰= 1
y'(0) = 1

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "4 Marks" from Differential Equations→Differential Equations — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Verify Lagrange's mean value theorem for the following function on the indicated intervals. Find a point 'c' in the indicated interval as stated by the Lagrange's mean value theorem.
f(x) = x²- 1 on [2, 3]

Show that the four points (0, -1, -1), (4, 5, 1), (3, 9, 4) and (-4, 4, 4) are coplanar and find the equation of the common plane.

If $\text{y}=\text{x}\sin(\text{a}+\text{y}),$ prove that $\frac{\text{dx}}{\text{dx}}=\frac{\sin^2(\text{a}+\text{y})}{\sin(\text{a}+\text{y})-\text{y}\cos(\text{a}+\text{y})}$

Differentiate the following functions with respect to x:
$\tan^{-1}\Big(\frac{\text{x}-\text{a}}{\text{x}+\text{a}}\Big)$

Find the vector equations of the following planes in scalar product form $(\vec{\text{r}}\cdot\vec{\text{n}}=\text{d}):$
$\vec{\text{r}}=(\hat{\text{i}}+\hat{\text{j}})+\lambda(\hat{\text{i}}+2\hat{\text{j}}-\hat{\text{k}})+\mu(-\hat{\text{i}}+\hat{\text{j}}-2\hat{\text{k}})$

Solve the following differential equation
$\frac{\text{dy}}{\text{dx}}=\frac{1+\text{y}^2}{\text{y}^3}$

Evaluate the following definite integrals:
$\int_{0}^\limits{1}\sqrt{\text{x}(1-\text{x})}\text{ dx}$

Find $\frac{\text{dy}}{\text{dx}},$ when
$\text{x}=\frac{3\text{at}}{1+\text{t}^2}\text{ and y}=\frac{3\text{at}^2}{1+\text{t}^2}$

Integrate the rational function $\frac{x}{\left(x^{2}+1\right)(x-1)}$

The volume of a cube increases at a constant rate. Prove that the increase in its surface area varies inversely as the length of the side.