General Solution of a Differential Equation

Q.

Which of the following is the solution of the differential equation $\frac{dy}{dx}=e^{y+x}$ with initial condition $y\left(0\right)=-\ln 4$?

1. $y=-x-\ln 4$

2. $y=x-\ln 4$

3. $y=-\ln \left(-e^x+5\right)$

4. $y=-\ln \left(e^x+3\right)$

5. $y=\ln \left(e^x+3\right)$

Q. General solution of a differential equation is $y=k{e}^x$. Find the particular solution if the curve passes through $(0,1)$

1. $y=k{e}^x$
2. $y=k{e}^x+C$
3. $y=e^x$
4. $y=2e^x$

Q. General solution of a differential equation is $y=k{e}^x$. Find the particular solution if the curve passes through $(0,1)$

1. $y=k{e}^x$
2. $y=k{e}^x+C$
3. $y=e^x$
4. $y=2e^x$

Q. Which of the following relation in x and y is general solution of the differential equation $\frac{dy}{dx}=y$

1. $y=e^x$
2. $y=e^x+c$
3. $y=k{e}^x$
4. $y=k{e}^x+c$

Q. The solution of $\left(y\right(1+x^{-1})+siny)dx+(x+logx+xcosy)dy=0$ is

1. $(1+y^{-1}siny)+x^{-1}logx=C$
   
2. $(y+siny)+xylogx=C$
   
3. $xy+ylogx+xsiny=C$
4. $Noneofthese$

Q. Solution of $2y\sin x\frac{dy}{dx}=2\sin x\cdot \cos x-y^2\cos x,x=\frac{\pi }{2},y=1$ is given by

1. $y^2=\sin x$
2. $y={\sin }^{2}x$
3. $y^2=\cos x+1$
4. None of these