(A) : Given, differential equation where centre lie on \(x\) axis- \(x^{2}+y^{2}-2 a x=0\) On differentiating both sides w.r.t. \(x\), we get- \(2 x+2 y \frac{d y}{d x}-2 a=0\) Again differentiating with respect to \(\mathrm{x}\), we get- \(y \frac{d^{2} y}{d x^{2}}+\left(\frac{d y}{d x}\right)^{2}=-1\) \(y \frac{d^{2} y}{d x^{2}}+\left(\frac{d y}{d x}\right)^{2}+1=0\)
MHT CET-2011
Differential Equation
87185
\(y=a \sin (\log x)+b \cos (\log x)\), then the differential equation without the parameter ' \(a\) ' \& ' \(b\) ' is
(B) : We have differential equation- \(\left(1+y^{2}\right) \tan ^{-1} x d x+\left(1+x^{2}\right) 2 y d y=0\) Separating the variable, we have - \(\frac{\tan ^{-1} x}{1+x^{2}} d x+\frac{2 y}{1+y^{2}} d y=0\) Integrating both side, we get- \(\int \frac{2 y}{1+y^{2}} d y=-\int \frac{\tan ^{-1} x}{1+x^{2}} d x\) \(\log \left|1+y^{2}\right|=-\frac{1}{2}\left(\tan ^{-1} x\right)^{2}+c\) \(\frac{1}{2}\left(\tan ^{-1} \mathrm{x}\right)^{2}+\log \left|1+\mathrm{y}^{2}\right|=\mathrm{c}\) \(\left(\tan ^{-1} \mathrm{x}\right)^{2}+2 \log \left|1+\mathrm{y}^{2}\right|=\mathrm{c}\)
MHT CET-2006
Differential Equation
87187
If \(y=\left(\tan ^{-1} x\right)^{2}\), then \(\frac{d^{2} y}{d x^{2}}\left(1+x^{2}\right)^{2}+2 x\left(x^{2}+1\right)\) \(\underline{\mathbf{d y}}=\)
(B) : Given that, \(y=\left(\tan ^{-1} x\right)^{2}\) Differentiating w.r.t \(\mathrm{x}\), we get - \(\frac{d y}{d x}=2 \tan ^{-1} x \cdot \frac{1}{1+x^{2}} \Rightarrow\left(1+x^{2}\right) \frac{d y}{d x}=2 \tan ^{-1} x\) Again differentiating w.r.t \(x\) we get- \(2 x \frac{d y}{d x}+\left(1+x^{2}\right) \frac{d^{2} y}{d x^{2}}=\frac{2}{1+x^{2}}\) \(2 x\left(1+x^{2}\right) \frac{d y}{d x}+\left(1+x^{2}\right)^{2} \frac{d^{2} y}{d x^{2}}=2\) \(\frac{d^{2} y}{d x^{2}}\left(1+x^{2}\right)^{2}+2 x\left(1+x^{2}\right) \frac{d y}{d x}=2\)
MHT CET-2006
Differential Equation
87188
The solution of the differential equation \(\frac{\mathbf{d y}}{\mathbf{d x}}=3^{x+y} \text { at } x=y=0 \text { is }\)
1 \(3^{\mathrm{x}}+3^{-\mathrm{y}}-2=0\)
2 \(3^{\mathrm{x}}-3^{-\mathrm{y}}-2=0\)
3 \(3^{\mathrm{x}}+3^{-\mathrm{y}}+\mathrm{c}=0\)
4 \(3^{\mathrm{x}}+3^{-\mathrm{y}}-\mathrm{c}=0\)
Explanation:
(A) : We have - \(\frac{d y}{d x}=3^{x+y} \Rightarrow \frac{d y}{d x}=3^{x} \cdot 3^{y} \Rightarrow \frac{d y}{3^{y}}=3^{x} d x\) Integrating on both side we get- \(\int 3^{-y} d y=\int 3^{x} d x \Rightarrow \frac{-3^{-y}}{\log 3}=\frac{3^{x}}{\log 3}+c\) \(3^{x}+3^{-y}=c\) When, \(x=y=0 \Rightarrow 3^{0}+3^{-0}=c \Rightarrow c=2\) Then, \(3^{\mathrm{x}}+3^{-\mathrm{y}}-2=0\)
(A) : Given, differential equation where centre lie on \(x\) axis- \(x^{2}+y^{2}-2 a x=0\) On differentiating both sides w.r.t. \(x\), we get- \(2 x+2 y \frac{d y}{d x}-2 a=0\) Again differentiating with respect to \(\mathrm{x}\), we get- \(y \frac{d^{2} y}{d x^{2}}+\left(\frac{d y}{d x}\right)^{2}=-1\) \(y \frac{d^{2} y}{d x^{2}}+\left(\frac{d y}{d x}\right)^{2}+1=0\)
MHT CET-2011
Differential Equation
87185
\(y=a \sin (\log x)+b \cos (\log x)\), then the differential equation without the parameter ' \(a\) ' \& ' \(b\) ' is
(B) : We have differential equation- \(\left(1+y^{2}\right) \tan ^{-1} x d x+\left(1+x^{2}\right) 2 y d y=0\) Separating the variable, we have - \(\frac{\tan ^{-1} x}{1+x^{2}} d x+\frac{2 y}{1+y^{2}} d y=0\) Integrating both side, we get- \(\int \frac{2 y}{1+y^{2}} d y=-\int \frac{\tan ^{-1} x}{1+x^{2}} d x\) \(\log \left|1+y^{2}\right|=-\frac{1}{2}\left(\tan ^{-1} x\right)^{2}+c\) \(\frac{1}{2}\left(\tan ^{-1} \mathrm{x}\right)^{2}+\log \left|1+\mathrm{y}^{2}\right|=\mathrm{c}\) \(\left(\tan ^{-1} \mathrm{x}\right)^{2}+2 \log \left|1+\mathrm{y}^{2}\right|=\mathrm{c}\)
MHT CET-2006
Differential Equation
87187
If \(y=\left(\tan ^{-1} x\right)^{2}\), then \(\frac{d^{2} y}{d x^{2}}\left(1+x^{2}\right)^{2}+2 x\left(x^{2}+1\right)\) \(\underline{\mathbf{d y}}=\)
(B) : Given that, \(y=\left(\tan ^{-1} x\right)^{2}\) Differentiating w.r.t \(\mathrm{x}\), we get - \(\frac{d y}{d x}=2 \tan ^{-1} x \cdot \frac{1}{1+x^{2}} \Rightarrow\left(1+x^{2}\right) \frac{d y}{d x}=2 \tan ^{-1} x\) Again differentiating w.r.t \(x\) we get- \(2 x \frac{d y}{d x}+\left(1+x^{2}\right) \frac{d^{2} y}{d x^{2}}=\frac{2}{1+x^{2}}\) \(2 x\left(1+x^{2}\right) \frac{d y}{d x}+\left(1+x^{2}\right)^{2} \frac{d^{2} y}{d x^{2}}=2\) \(\frac{d^{2} y}{d x^{2}}\left(1+x^{2}\right)^{2}+2 x\left(1+x^{2}\right) \frac{d y}{d x}=2\)
MHT CET-2006
Differential Equation
87188
The solution of the differential equation \(\frac{\mathbf{d y}}{\mathbf{d x}}=3^{x+y} \text { at } x=y=0 \text { is }\)
1 \(3^{\mathrm{x}}+3^{-\mathrm{y}}-2=0\)
2 \(3^{\mathrm{x}}-3^{-\mathrm{y}}-2=0\)
3 \(3^{\mathrm{x}}+3^{-\mathrm{y}}+\mathrm{c}=0\)
4 \(3^{\mathrm{x}}+3^{-\mathrm{y}}-\mathrm{c}=0\)
Explanation:
(A) : We have - \(\frac{d y}{d x}=3^{x+y} \Rightarrow \frac{d y}{d x}=3^{x} \cdot 3^{y} \Rightarrow \frac{d y}{3^{y}}=3^{x} d x\) Integrating on both side we get- \(\int 3^{-y} d y=\int 3^{x} d x \Rightarrow \frac{-3^{-y}}{\log 3}=\frac{3^{x}}{\log 3}+c\) \(3^{x}+3^{-y}=c\) When, \(x=y=0 \Rightarrow 3^{0}+3^{-0}=c \Rightarrow c=2\) Then, \(3^{\mathrm{x}}+3^{-\mathrm{y}}-2=0\)
(A) : Given, differential equation where centre lie on \(x\) axis- \(x^{2}+y^{2}-2 a x=0\) On differentiating both sides w.r.t. \(x\), we get- \(2 x+2 y \frac{d y}{d x}-2 a=0\) Again differentiating with respect to \(\mathrm{x}\), we get- \(y \frac{d^{2} y}{d x^{2}}+\left(\frac{d y}{d x}\right)^{2}=-1\) \(y \frac{d^{2} y}{d x^{2}}+\left(\frac{d y}{d x}\right)^{2}+1=0\)
MHT CET-2011
Differential Equation
87185
\(y=a \sin (\log x)+b \cos (\log x)\), then the differential equation without the parameter ' \(a\) ' \& ' \(b\) ' is
(B) : We have differential equation- \(\left(1+y^{2}\right) \tan ^{-1} x d x+\left(1+x^{2}\right) 2 y d y=0\) Separating the variable, we have - \(\frac{\tan ^{-1} x}{1+x^{2}} d x+\frac{2 y}{1+y^{2}} d y=0\) Integrating both side, we get- \(\int \frac{2 y}{1+y^{2}} d y=-\int \frac{\tan ^{-1} x}{1+x^{2}} d x\) \(\log \left|1+y^{2}\right|=-\frac{1}{2}\left(\tan ^{-1} x\right)^{2}+c\) \(\frac{1}{2}\left(\tan ^{-1} \mathrm{x}\right)^{2}+\log \left|1+\mathrm{y}^{2}\right|=\mathrm{c}\) \(\left(\tan ^{-1} \mathrm{x}\right)^{2}+2 \log \left|1+\mathrm{y}^{2}\right|=\mathrm{c}\)
MHT CET-2006
Differential Equation
87187
If \(y=\left(\tan ^{-1} x\right)^{2}\), then \(\frac{d^{2} y}{d x^{2}}\left(1+x^{2}\right)^{2}+2 x\left(x^{2}+1\right)\) \(\underline{\mathbf{d y}}=\)
(B) : Given that, \(y=\left(\tan ^{-1} x\right)^{2}\) Differentiating w.r.t \(\mathrm{x}\), we get - \(\frac{d y}{d x}=2 \tan ^{-1} x \cdot \frac{1}{1+x^{2}} \Rightarrow\left(1+x^{2}\right) \frac{d y}{d x}=2 \tan ^{-1} x\) Again differentiating w.r.t \(x\) we get- \(2 x \frac{d y}{d x}+\left(1+x^{2}\right) \frac{d^{2} y}{d x^{2}}=\frac{2}{1+x^{2}}\) \(2 x\left(1+x^{2}\right) \frac{d y}{d x}+\left(1+x^{2}\right)^{2} \frac{d^{2} y}{d x^{2}}=2\) \(\frac{d^{2} y}{d x^{2}}\left(1+x^{2}\right)^{2}+2 x\left(1+x^{2}\right) \frac{d y}{d x}=2\)
MHT CET-2006
Differential Equation
87188
The solution of the differential equation \(\frac{\mathbf{d y}}{\mathbf{d x}}=3^{x+y} \text { at } x=y=0 \text { is }\)
1 \(3^{\mathrm{x}}+3^{-\mathrm{y}}-2=0\)
2 \(3^{\mathrm{x}}-3^{-\mathrm{y}}-2=0\)
3 \(3^{\mathrm{x}}+3^{-\mathrm{y}}+\mathrm{c}=0\)
4 \(3^{\mathrm{x}}+3^{-\mathrm{y}}-\mathrm{c}=0\)
Explanation:
(A) : We have - \(\frac{d y}{d x}=3^{x+y} \Rightarrow \frac{d y}{d x}=3^{x} \cdot 3^{y} \Rightarrow \frac{d y}{3^{y}}=3^{x} d x\) Integrating on both side we get- \(\int 3^{-y} d y=\int 3^{x} d x \Rightarrow \frac{-3^{-y}}{\log 3}=\frac{3^{x}}{\log 3}+c\) \(3^{x}+3^{-y}=c\) When, \(x=y=0 \Rightarrow 3^{0}+3^{-0}=c \Rightarrow c=2\) Then, \(3^{\mathrm{x}}+3^{-\mathrm{y}}-2=0\)
(A) : Given, differential equation where centre lie on \(x\) axis- \(x^{2}+y^{2}-2 a x=0\) On differentiating both sides w.r.t. \(x\), we get- \(2 x+2 y \frac{d y}{d x}-2 a=0\) Again differentiating with respect to \(\mathrm{x}\), we get- \(y \frac{d^{2} y}{d x^{2}}+\left(\frac{d y}{d x}\right)^{2}=-1\) \(y \frac{d^{2} y}{d x^{2}}+\left(\frac{d y}{d x}\right)^{2}+1=0\)
MHT CET-2011
Differential Equation
87185
\(y=a \sin (\log x)+b \cos (\log x)\), then the differential equation without the parameter ' \(a\) ' \& ' \(b\) ' is
(B) : We have differential equation- \(\left(1+y^{2}\right) \tan ^{-1} x d x+\left(1+x^{2}\right) 2 y d y=0\) Separating the variable, we have - \(\frac{\tan ^{-1} x}{1+x^{2}} d x+\frac{2 y}{1+y^{2}} d y=0\) Integrating both side, we get- \(\int \frac{2 y}{1+y^{2}} d y=-\int \frac{\tan ^{-1} x}{1+x^{2}} d x\) \(\log \left|1+y^{2}\right|=-\frac{1}{2}\left(\tan ^{-1} x\right)^{2}+c\) \(\frac{1}{2}\left(\tan ^{-1} \mathrm{x}\right)^{2}+\log \left|1+\mathrm{y}^{2}\right|=\mathrm{c}\) \(\left(\tan ^{-1} \mathrm{x}\right)^{2}+2 \log \left|1+\mathrm{y}^{2}\right|=\mathrm{c}\)
MHT CET-2006
Differential Equation
87187
If \(y=\left(\tan ^{-1} x\right)^{2}\), then \(\frac{d^{2} y}{d x^{2}}\left(1+x^{2}\right)^{2}+2 x\left(x^{2}+1\right)\) \(\underline{\mathbf{d y}}=\)
(B) : Given that, \(y=\left(\tan ^{-1} x\right)^{2}\) Differentiating w.r.t \(\mathrm{x}\), we get - \(\frac{d y}{d x}=2 \tan ^{-1} x \cdot \frac{1}{1+x^{2}} \Rightarrow\left(1+x^{2}\right) \frac{d y}{d x}=2 \tan ^{-1} x\) Again differentiating w.r.t \(x\) we get- \(2 x \frac{d y}{d x}+\left(1+x^{2}\right) \frac{d^{2} y}{d x^{2}}=\frac{2}{1+x^{2}}\) \(2 x\left(1+x^{2}\right) \frac{d y}{d x}+\left(1+x^{2}\right)^{2} \frac{d^{2} y}{d x^{2}}=2\) \(\frac{d^{2} y}{d x^{2}}\left(1+x^{2}\right)^{2}+2 x\left(1+x^{2}\right) \frac{d y}{d x}=2\)
MHT CET-2006
Differential Equation
87188
The solution of the differential equation \(\frac{\mathbf{d y}}{\mathbf{d x}}=3^{x+y} \text { at } x=y=0 \text { is }\)
1 \(3^{\mathrm{x}}+3^{-\mathrm{y}}-2=0\)
2 \(3^{\mathrm{x}}-3^{-\mathrm{y}}-2=0\)
3 \(3^{\mathrm{x}}+3^{-\mathrm{y}}+\mathrm{c}=0\)
4 \(3^{\mathrm{x}}+3^{-\mathrm{y}}-\mathrm{c}=0\)
Explanation:
(A) : We have - \(\frac{d y}{d x}=3^{x+y} \Rightarrow \frac{d y}{d x}=3^{x} \cdot 3^{y} \Rightarrow \frac{d y}{3^{y}}=3^{x} d x\) Integrating on both side we get- \(\int 3^{-y} d y=\int 3^{x} d x \Rightarrow \frac{-3^{-y}}{\log 3}=\frac{3^{x}}{\log 3}+c\) \(3^{x}+3^{-y}=c\) When, \(x=y=0 \Rightarrow 3^{0}+3^{-0}=c \Rightarrow c=2\) Then, \(3^{\mathrm{x}}+3^{-\mathrm{y}}-2=0\)
(A) : Given, differential equation where centre lie on \(x\) axis- \(x^{2}+y^{2}-2 a x=0\) On differentiating both sides w.r.t. \(x\), we get- \(2 x+2 y \frac{d y}{d x}-2 a=0\) Again differentiating with respect to \(\mathrm{x}\), we get- \(y \frac{d^{2} y}{d x^{2}}+\left(\frac{d y}{d x}\right)^{2}=-1\) \(y \frac{d^{2} y}{d x^{2}}+\left(\frac{d y}{d x}\right)^{2}+1=0\)
MHT CET-2011
Differential Equation
87185
\(y=a \sin (\log x)+b \cos (\log x)\), then the differential equation without the parameter ' \(a\) ' \& ' \(b\) ' is
(B) : We have differential equation- \(\left(1+y^{2}\right) \tan ^{-1} x d x+\left(1+x^{2}\right) 2 y d y=0\) Separating the variable, we have - \(\frac{\tan ^{-1} x}{1+x^{2}} d x+\frac{2 y}{1+y^{2}} d y=0\) Integrating both side, we get- \(\int \frac{2 y}{1+y^{2}} d y=-\int \frac{\tan ^{-1} x}{1+x^{2}} d x\) \(\log \left|1+y^{2}\right|=-\frac{1}{2}\left(\tan ^{-1} x\right)^{2}+c\) \(\frac{1}{2}\left(\tan ^{-1} \mathrm{x}\right)^{2}+\log \left|1+\mathrm{y}^{2}\right|=\mathrm{c}\) \(\left(\tan ^{-1} \mathrm{x}\right)^{2}+2 \log \left|1+\mathrm{y}^{2}\right|=\mathrm{c}\)
MHT CET-2006
Differential Equation
87187
If \(y=\left(\tan ^{-1} x\right)^{2}\), then \(\frac{d^{2} y}{d x^{2}}\left(1+x^{2}\right)^{2}+2 x\left(x^{2}+1\right)\) \(\underline{\mathbf{d y}}=\)
(B) : Given that, \(y=\left(\tan ^{-1} x\right)^{2}\) Differentiating w.r.t \(\mathrm{x}\), we get - \(\frac{d y}{d x}=2 \tan ^{-1} x \cdot \frac{1}{1+x^{2}} \Rightarrow\left(1+x^{2}\right) \frac{d y}{d x}=2 \tan ^{-1} x\) Again differentiating w.r.t \(x\) we get- \(2 x \frac{d y}{d x}+\left(1+x^{2}\right) \frac{d^{2} y}{d x^{2}}=\frac{2}{1+x^{2}}\) \(2 x\left(1+x^{2}\right) \frac{d y}{d x}+\left(1+x^{2}\right)^{2} \frac{d^{2} y}{d x^{2}}=2\) \(\frac{d^{2} y}{d x^{2}}\left(1+x^{2}\right)^{2}+2 x\left(1+x^{2}\right) \frac{d y}{d x}=2\)
MHT CET-2006
Differential Equation
87188
The solution of the differential equation \(\frac{\mathbf{d y}}{\mathbf{d x}}=3^{x+y} \text { at } x=y=0 \text { is }\)
1 \(3^{\mathrm{x}}+3^{-\mathrm{y}}-2=0\)
2 \(3^{\mathrm{x}}-3^{-\mathrm{y}}-2=0\)
3 \(3^{\mathrm{x}}+3^{-\mathrm{y}}+\mathrm{c}=0\)
4 \(3^{\mathrm{x}}+3^{-\mathrm{y}}-\mathrm{c}=0\)
Explanation:
(A) : We have - \(\frac{d y}{d x}=3^{x+y} \Rightarrow \frac{d y}{d x}=3^{x} \cdot 3^{y} \Rightarrow \frac{d y}{3^{y}}=3^{x} d x\) Integrating on both side we get- \(\int 3^{-y} d y=\int 3^{x} d x \Rightarrow \frac{-3^{-y}}{\log 3}=\frac{3^{x}}{\log 3}+c\) \(3^{x}+3^{-y}=c\) When, \(x=y=0 \Rightarrow 3^{0}+3^{-0}=c \Rightarrow c=2\) Then, \(3^{\mathrm{x}}+3^{-\mathrm{y}}-2=0\)
