Question
Write the solution of the differential equation $\frac{d y}{d x}=2^{-y}$
✓
Answer
Given differential equation is
$\frac{d y}{d x}=2^{-y}$
On separating the variables, we get
$2^y d y=d x$
On integrating both sides, we get c
$\int 2^y d y=\int d x$
or $\frac{2^y}{\log 2}=x+C_1$
or $2^y=x \log 2+C_1 \log 2$
$\therefore 2^y=x \log 2+C$, where $C=C_1 \log 2$
$\frac{d y}{d x}=2^{-y}$
On separating the variables, we get
$2^y d y=d x$
On integrating both sides, we get c
$\int 2^y d y=\int d x$
or $\frac{2^y}{\log 2}=x+C_1$
or $2^y=x \log 2+C_1 \log 2$
$\therefore 2^y=x \log 2+C$, where $C=C_1 \log 2$
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Mohan is the student of class XII of Sunshine School. In the mathematics class, teacher defines the method for evaluation of definite integral, she said, To evaluate the definite integral $\underset{\pi}{\equiv} f(x) d x$ a continuous function f(x) defined on [a, b], we may use the following algorithm.
Step 1: Find the indefinite integral $\stackrel{b}{\equiv} f(x) d x$
Let this be P(x) There is no need to keep the constant of integration.
Step 2: Evaluate $\phi(b)$ and $\phi(a)$.
Step 3: Calculate $\phi(b)-\phi(a)$.
The number obtained in step 3 is the value of definite integral $\int_a^b f(x) d x$
Q.1. Evaluate: $\int_0^1 \frac{1}{\sqrt{1+x}+\sqrt{x}} d x$
Q.2. If $\int_1^a\left(3 x^2+2 x+1\right) d x=1$, find real values of $a$.
→Mohan is the student of class XII of Sunshine School. In the mathematics class, teacher defines the method for evaluation of definite integral, she said, To evaluate the definite integral $\underset{\pi}{\equiv} f(x) d x$ a continuous function f(x) defined on [a, b], we may use the following algorithm.
Step 1: Find the indefinite integral $\stackrel{b}{\equiv} f(x) d x$
Let this be P(x) There is no need to keep the constant of integration.
Step 2: Evaluate $\phi(b)$ and $\phi(a)$.
Step 3: Calculate $\phi(b)-\phi(a)$.
The number obtained in step 3 is the value of definite integral $\int_a^b f(x) d x$
Q.1. Evaluate: $\int_0^1 \frac{1}{\sqrt{1+x}+\sqrt{x}} d x$
Q.2. If $\int_1^a\left(3 x^2+2 x+1\right) d x=1$, find real values of $a$.