The solution of differential equation. day/dx = 1/√1+x^2 is y =?
- sinh^-1 x+c
- sech^-1x+c
- tanh^-1x+c
- cotx+c
Explanation
The given differential equation is:
dy/dx = 1/√(1+x^2)
To solve this equation, we can use the substitution u = 1+x^2, which implies du/dx = 2x. Then, we can rewrite the equation as:
dy/dx = 1/√u
dy = 1/√u du/2x
2y = ∫1/√u du
Now, we can recognize the integral as the inverse hyperbolic sine (arsinh or sinh^-1) function:
2y = sinh^-1(u) + C
2y = sinh^-1(1+x^2) + C
Dividing both sides by 2, we get:
y = (1/2)sinh^-1(1+x^2) + C/2
Since C/2 is also a constant, we can write the solution as:
y = (1/2)sinh^-1(1+x^2) + c
or simply: y = sinh^-1(x) + c
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