alleryThe number of arbitrary constants in the general solution of a differential equation of order $n$ is:
A$n$
B$0$
C$1$
D$n - 1$
Answer & Solution
Correct answer: A. $n$
By definition, the general solution of an $n$-th order differential equation contains exactly $n$ independent arbitrary constants, matching the order of the equation.
For example, $\dfrac{d^2y}{dx^2} + y = 0$ has the general solution $y = A\cos x + B\sin x$ with two constants $A$ and $B$, since the equation is of order 2.
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The integrating factor of the linear differential equation $\dfrac{dy}{dx} + \dfrac{y}{x} Using the substitution $t = 1 + x^2$, $\displaystyle\int \dfrac{x}{1 + x^2}\,dx$ equals:$\displaystyle\int e^x\bigl(f(x) + f'(x)\bigr)\,dx$ equals:$\displaystyle\int \log x\,dx$ equals:$\displaystyle\int ec^2 x\,dx$ equals:$\displaystyle\int \dfrac{1}{ qrt{1 - x^2}}\,dx$ equals:$\displaystyle\int \dfrac{1}{1+x^2}\,dx$ equals:$\displaystyle\int \dfrac{1}{x^2 - a^2}\,dx$ (with $|x| > a > 0$) equals: