87203
Solution of differential equation \(\mathbf{x d y} \mathbf{-} \mathbf{y d x}=0\) represents
1 a rectangular hyperbola
2 parabola whose vertex is at origin
3 straight line passing through origin
4 a circle whose centre is origin
Explanation:
(C) :We have differential equation- \(x d y-y d x=0\) \(x d y=y d x\) \(\frac{\mathrm{dy}}{\mathrm{y}}=\frac{\mathrm{dx}}{\mathrm{x}}\) Integrating both side, we get- \(\int \frac{d y}{y}=\int \frac{d x}{x}\) \(\log \mathrm{y}=\log \mathrm{x}+\log \mathrm{c}\) \(\mathrm{y}=\mathrm{cx}\) Which is a straight line passing through the origin.
Karnataka CET-2021
Differential Equation
87204
The solution of \(\frac{d y}{d x}-1=e^{x-y}\) is
(D) : Given that the differential equation - \(\frac{d y}{d x}-1=e^{x-y} \Rightarrow \frac{d y}{d x}-1=\frac{e^{x}}{e^{y}}\) \(e^{y} d y-e^{y} d x=e^{x} d x\) \(\mathrm{e}^{\mathrm{y}-\mathrm{x}}(\mathrm{dy}-\mathrm{dx})=\mathrm{dx}\) Integrating both sides, \(\int e^{y-x}(d y-d x)=\int d x+c\) Put \(\mathrm{y}-\mathrm{x}=\mathrm{t}\) \(\mathrm{d}(\mathrm{y}-\mathrm{x})=\mathrm{dt}\) \(\therefore \int \mathrm{e}^{\mathrm{t}} \mathrm{dt}=\mathrm{x}+\mathrm{c}\) \(\mathrm{e}^{\mathrm{y}-\mathrm{x}}=\mathrm{x}+\mathrm{c}\)
COMEDK-2011
Differential Equation
87239
The solution of the differential equation \(\left(x^{2}-y x^{2}\right) \frac{d y}{d x}+y^{2}+x y^{2}=0\) is
(A) : Given, \(\left(x^{2}-y x^{2}\right) \frac{d y}{d x}+y^{2}+x y^{2}=0\) \(x^{2}(1-y) d y+\left(y^{2}+x y^{2}\right) d x=0\) Separating the variable, we get- \(\frac{(1-y) d y}{y^{2}}+\frac{(1+x) d x}{x^{2}}=0\) \(\left(\frac{1}{y^{2}}-\frac{1}{y}\right) d y+\left(\frac{1}{x^{2}}+\frac{1}{x}\right) d x=0\) Integrating both side, we get- \(-\frac{1}{y}-\log y-\frac{1}{x}+\log x=\log C\) \(\log x-\log y=\frac{1}{x}+\frac{1}{y}+C\) \(\log \left(\frac{\mathrm{x}}{\mathrm{y}}\right)=\frac{1}{\mathrm{x}}+\frac{1}{\mathrm{y}}+\mathrm{C}\)
JCECE-2014
Differential Equation
87240
Find the differential equation of curves \(\mathbf{y}=\mathbf{A} \mathbf{e}^{\mathbf{x}}+\mathbf{B e}^{-\mathbf{x}}\) for different values of \(A\) and \(B\)
87203
Solution of differential equation \(\mathbf{x d y} \mathbf{-} \mathbf{y d x}=0\) represents
1 a rectangular hyperbola
2 parabola whose vertex is at origin
3 straight line passing through origin
4 a circle whose centre is origin
Explanation:
(C) :We have differential equation- \(x d y-y d x=0\) \(x d y=y d x\) \(\frac{\mathrm{dy}}{\mathrm{y}}=\frac{\mathrm{dx}}{\mathrm{x}}\) Integrating both side, we get- \(\int \frac{d y}{y}=\int \frac{d x}{x}\) \(\log \mathrm{y}=\log \mathrm{x}+\log \mathrm{c}\) \(\mathrm{y}=\mathrm{cx}\) Which is a straight line passing through the origin.
Karnataka CET-2021
Differential Equation
87204
The solution of \(\frac{d y}{d x}-1=e^{x-y}\) is
(D) : Given that the differential equation - \(\frac{d y}{d x}-1=e^{x-y} \Rightarrow \frac{d y}{d x}-1=\frac{e^{x}}{e^{y}}\) \(e^{y} d y-e^{y} d x=e^{x} d x\) \(\mathrm{e}^{\mathrm{y}-\mathrm{x}}(\mathrm{dy}-\mathrm{dx})=\mathrm{dx}\) Integrating both sides, \(\int e^{y-x}(d y-d x)=\int d x+c\) Put \(\mathrm{y}-\mathrm{x}=\mathrm{t}\) \(\mathrm{d}(\mathrm{y}-\mathrm{x})=\mathrm{dt}\) \(\therefore \int \mathrm{e}^{\mathrm{t}} \mathrm{dt}=\mathrm{x}+\mathrm{c}\) \(\mathrm{e}^{\mathrm{y}-\mathrm{x}}=\mathrm{x}+\mathrm{c}\)
COMEDK-2011
Differential Equation
87239
The solution of the differential equation \(\left(x^{2}-y x^{2}\right) \frac{d y}{d x}+y^{2}+x y^{2}=0\) is
(A) : Given, \(\left(x^{2}-y x^{2}\right) \frac{d y}{d x}+y^{2}+x y^{2}=0\) \(x^{2}(1-y) d y+\left(y^{2}+x y^{2}\right) d x=0\) Separating the variable, we get- \(\frac{(1-y) d y}{y^{2}}+\frac{(1+x) d x}{x^{2}}=0\) \(\left(\frac{1}{y^{2}}-\frac{1}{y}\right) d y+\left(\frac{1}{x^{2}}+\frac{1}{x}\right) d x=0\) Integrating both side, we get- \(-\frac{1}{y}-\log y-\frac{1}{x}+\log x=\log C\) \(\log x-\log y=\frac{1}{x}+\frac{1}{y}+C\) \(\log \left(\frac{\mathrm{x}}{\mathrm{y}}\right)=\frac{1}{\mathrm{x}}+\frac{1}{\mathrm{y}}+\mathrm{C}\)
JCECE-2014
Differential Equation
87240
Find the differential equation of curves \(\mathbf{y}=\mathbf{A} \mathbf{e}^{\mathbf{x}}+\mathbf{B e}^{-\mathbf{x}}\) for different values of \(A\) and \(B\)
87203
Solution of differential equation \(\mathbf{x d y} \mathbf{-} \mathbf{y d x}=0\) represents
1 a rectangular hyperbola
2 parabola whose vertex is at origin
3 straight line passing through origin
4 a circle whose centre is origin
Explanation:
(C) :We have differential equation- \(x d y-y d x=0\) \(x d y=y d x\) \(\frac{\mathrm{dy}}{\mathrm{y}}=\frac{\mathrm{dx}}{\mathrm{x}}\) Integrating both side, we get- \(\int \frac{d y}{y}=\int \frac{d x}{x}\) \(\log \mathrm{y}=\log \mathrm{x}+\log \mathrm{c}\) \(\mathrm{y}=\mathrm{cx}\) Which is a straight line passing through the origin.
Karnataka CET-2021
Differential Equation
87204
The solution of \(\frac{d y}{d x}-1=e^{x-y}\) is
(D) : Given that the differential equation - \(\frac{d y}{d x}-1=e^{x-y} \Rightarrow \frac{d y}{d x}-1=\frac{e^{x}}{e^{y}}\) \(e^{y} d y-e^{y} d x=e^{x} d x\) \(\mathrm{e}^{\mathrm{y}-\mathrm{x}}(\mathrm{dy}-\mathrm{dx})=\mathrm{dx}\) Integrating both sides, \(\int e^{y-x}(d y-d x)=\int d x+c\) Put \(\mathrm{y}-\mathrm{x}=\mathrm{t}\) \(\mathrm{d}(\mathrm{y}-\mathrm{x})=\mathrm{dt}\) \(\therefore \int \mathrm{e}^{\mathrm{t}} \mathrm{dt}=\mathrm{x}+\mathrm{c}\) \(\mathrm{e}^{\mathrm{y}-\mathrm{x}}=\mathrm{x}+\mathrm{c}\)
COMEDK-2011
Differential Equation
87239
The solution of the differential equation \(\left(x^{2}-y x^{2}\right) \frac{d y}{d x}+y^{2}+x y^{2}=0\) is
(A) : Given, \(\left(x^{2}-y x^{2}\right) \frac{d y}{d x}+y^{2}+x y^{2}=0\) \(x^{2}(1-y) d y+\left(y^{2}+x y^{2}\right) d x=0\) Separating the variable, we get- \(\frac{(1-y) d y}{y^{2}}+\frac{(1+x) d x}{x^{2}}=0\) \(\left(\frac{1}{y^{2}}-\frac{1}{y}\right) d y+\left(\frac{1}{x^{2}}+\frac{1}{x}\right) d x=0\) Integrating both side, we get- \(-\frac{1}{y}-\log y-\frac{1}{x}+\log x=\log C\) \(\log x-\log y=\frac{1}{x}+\frac{1}{y}+C\) \(\log \left(\frac{\mathrm{x}}{\mathrm{y}}\right)=\frac{1}{\mathrm{x}}+\frac{1}{\mathrm{y}}+\mathrm{C}\)
JCECE-2014
Differential Equation
87240
Find the differential equation of curves \(\mathbf{y}=\mathbf{A} \mathbf{e}^{\mathbf{x}}+\mathbf{B e}^{-\mathbf{x}}\) for different values of \(A\) and \(B\)
87203
Solution of differential equation \(\mathbf{x d y} \mathbf{-} \mathbf{y d x}=0\) represents
1 a rectangular hyperbola
2 parabola whose vertex is at origin
3 straight line passing through origin
4 a circle whose centre is origin
Explanation:
(C) :We have differential equation- \(x d y-y d x=0\) \(x d y=y d x\) \(\frac{\mathrm{dy}}{\mathrm{y}}=\frac{\mathrm{dx}}{\mathrm{x}}\) Integrating both side, we get- \(\int \frac{d y}{y}=\int \frac{d x}{x}\) \(\log \mathrm{y}=\log \mathrm{x}+\log \mathrm{c}\) \(\mathrm{y}=\mathrm{cx}\) Which is a straight line passing through the origin.
Karnataka CET-2021
Differential Equation
87204
The solution of \(\frac{d y}{d x}-1=e^{x-y}\) is
(D) : Given that the differential equation - \(\frac{d y}{d x}-1=e^{x-y} \Rightarrow \frac{d y}{d x}-1=\frac{e^{x}}{e^{y}}\) \(e^{y} d y-e^{y} d x=e^{x} d x\) \(\mathrm{e}^{\mathrm{y}-\mathrm{x}}(\mathrm{dy}-\mathrm{dx})=\mathrm{dx}\) Integrating both sides, \(\int e^{y-x}(d y-d x)=\int d x+c\) Put \(\mathrm{y}-\mathrm{x}=\mathrm{t}\) \(\mathrm{d}(\mathrm{y}-\mathrm{x})=\mathrm{dt}\) \(\therefore \int \mathrm{e}^{\mathrm{t}} \mathrm{dt}=\mathrm{x}+\mathrm{c}\) \(\mathrm{e}^{\mathrm{y}-\mathrm{x}}=\mathrm{x}+\mathrm{c}\)
COMEDK-2011
Differential Equation
87239
The solution of the differential equation \(\left(x^{2}-y x^{2}\right) \frac{d y}{d x}+y^{2}+x y^{2}=0\) is
(A) : Given, \(\left(x^{2}-y x^{2}\right) \frac{d y}{d x}+y^{2}+x y^{2}=0\) \(x^{2}(1-y) d y+\left(y^{2}+x y^{2}\right) d x=0\) Separating the variable, we get- \(\frac{(1-y) d y}{y^{2}}+\frac{(1+x) d x}{x^{2}}=0\) \(\left(\frac{1}{y^{2}}-\frac{1}{y}\right) d y+\left(\frac{1}{x^{2}}+\frac{1}{x}\right) d x=0\) Integrating both side, we get- \(-\frac{1}{y}-\log y-\frac{1}{x}+\log x=\log C\) \(\log x-\log y=\frac{1}{x}+\frac{1}{y}+C\) \(\log \left(\frac{\mathrm{x}}{\mathrm{y}}\right)=\frac{1}{\mathrm{x}}+\frac{1}{\mathrm{y}}+\mathrm{C}\)
JCECE-2014
Differential Equation
87240
Find the differential equation of curves \(\mathbf{y}=\mathbf{A} \mathbf{e}^{\mathbf{x}}+\mathbf{B e}^{-\mathbf{x}}\) for different values of \(A\) and \(B\)
