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Differential Equation

87566 The curve, for which the area of the triangle formed by \(\mathrm{X}\)-axis, the tangent line at any point \(P\) and line OP is equal to \(a^{2}\), is given by

1 \(y=x-C x^{2}\)

2 \(x=C y \pm \frac{a^{2}}{y}\)

3 \(y=C x \pm \frac{a^{2}}{x}\)

4 None of these

Manipal UGET-2020]**#

Differential Equation

87567 The general solution of the differential equation
\(x \cos \frac{y}{x}(y d x+x d y)=y \sin \frac{y}{x}(x d y-y d x)\) is

1 \(\log (x y)=\log \cos \frac{x}{y}+C\)

2 \(\cos \left(\frac{y}{x}\right)=\frac{C}{x y}\)

3 \(\log (x y)=\log \sec \frac{x}{y}+C\)

4 \(x+y+C=0\)

Explanation:

(B) We have,
\(x \cos \left(\frac{y}{x}\right)(y d x+x d y)\)
\(=y \sin \frac{y}{x}(x d y-y d x)\)
\(\Rightarrow x y \sin \left(\frac{y}{x}\right) d y-y^{2} \sin \left(\frac{y}{x}\right) d x\)
\(=x y \cos \left(\frac{y}{x}\right) d x+x^{2} \cos \left(\frac{y}{x}\right) d y\)
\(\Rightarrow\left[x y \sin \left(\frac{y}{x}\right)-x^{2} \cos \left(\frac{y}{x}\right)\right] d y\)
\(=\left[x y \cos \left(\frac{y}{x}\right)+y^{2} \sin \left(\frac{y}{x}\right)\right] d x\)
\(\Rightarrow \frac{d y}{d x} =\frac{x y \cos \left(\frac{y}{x}\right)+y^{2} \sin \left(\frac{y}{x}\right)}{x y \sin \left(\frac{y}{x}\right)-x^{2} \cos \left(\frac{y}{x}\right)}\)
\(\Rightarrow \frac{d y}{d x} =\frac{\frac{y}{x} \cos \left(\frac{y}{x}\right)+\left(\frac{y}{x}\right)^{2} \sin \left(\frac{y}{x}\right)}{\frac{y}{x} \sin \left(\frac{y}{x}\right)-\cos \left(\frac{y}{x}\right)}\)
Put, \(y=x v \Rightarrow \frac{d y}{d x}=v+x \frac{d v}{d x}\)
\(\therefore \mathrm{v}+\mathrm{x} \frac{\mathrm{dv}}{\mathrm{dx}}=\frac{\mathrm{v} \cos \mathrm{v}+\mathrm{v}^{2} \sin \mathrm{v}}{\mathrm{v} \sin \mathrm{v}-\cos \mathrm{v}}\)
\(\Rightarrow \mathrm{x} \frac{\mathrm{dv}}{\mathrm{dx}}=\frac{2 \mathrm{v} \cos \mathrm{v}}{\mathrm{v} \sin \mathrm{v}-\cos \mathrm{v}} \Rightarrow \frac{\mathrm{v} \sin \mathrm{v}-\cos \mathrm{v}}{\mathrm{v} \cos \mathrm{v}} \mathrm{dv}=2 \frac{\mathrm{dx}}{\mathrm{x}}\)
\(\Rightarrow \int\left(\tan \mathrm{v}-\frac{1}{\mathrm{v}}\right) \mathrm{dv}=2 \int \frac{\mathrm{dx}}{\mathrm{x}}\)
\(\Rightarrow \log |\sec \mathrm{v}|-\log |\mathrm{v}|=2 \log |\mathrm{x}|+\log \left|\mathrm{C}_{1}\right|\)
\(\Rightarrow \log \left|\frac{\sec \mathrm{v}}{\mathrm{v}}\right|-\log |\mathrm{x}|^{2}=\log \left|\mathrm{C}_{1}\right|\)
\(\Rightarrow \log \left|\frac{\sec v}{v x^{2}}\right|=C_{1} \Rightarrow \frac{\sec \left(\frac{y}{x}\right)}{\frac{y}{x}\left(x^{2}\right)}=C_{1}\)
\(\Rightarrow \frac{\sec \left(\frac{y}{x}\right)}{x y}=C_{1} \Rightarrow \sec \left(\frac{y}{x}\right)=C_{1} x y\)
\(\Rightarrow \cos \left(\frac{\mathrm{y}}{\mathrm{x}}\right)=\frac{\mathrm{C}}{\mathrm{xy}}\)
\(\left(\because\right.\) where \(\left.\frac{1}{\mathrm{C}_{1}}=\mathrm{C}\right)\)

TS EAMCET-10.09.2020

Differential Equation

87568 Solution of the differential equation \(y^{\prime}=\) \(\frac{\left(x^{2}+y^{2}\right)}{x y}, y(1)=-2\) is given by

1 \(y_{2}^{2}=x^{2} \log x^{2}+4 x^{2}\)

2 \(y_{2}^{2}=x^{2} \log x-x^{2}\)

3 \(y^{2} x \log x^{2}+4 x^{2}\)

4 \(y^2=4 x^2 \log x^2+x^2\)

MHT CET-2021

Differential Equation

87569 The solution of the differential equation \(x \frac{d y}{d x}+y=\frac{1}{x^{2}}\) at \((1,2)\) is

1 \(x^{2} y+1=3 x\)

2 \(x^{2} y+1=0\)

3 \(x y+1=3 x\)

4 \(x^{2}(y+1)=3 x\)

5 \(x^{2} y=3 x+1\)

Kerala CEE-2011

Differential Equation

87570 The general solution of the differential equation \(\frac{d y}{d x}=e^{y}\left(e^{x}+e^{-x}+2 x\right)\) is

1 \(\mathrm{e}^{-\mathrm{y}}=\mathrm{e}^{\mathrm{x}}-\mathrm{e}^{-\mathrm{x}}+\mathrm{x}^{2}+\mathrm{C}\)

2 \(\mathrm{e}^{-\mathrm{y}}=\mathrm{e}^{-\mathrm{x}}-\mathrm{e}^{\mathrm{x}}-\mathrm{x}^{2}+\mathrm{C}\)

3 \(\mathrm{e}^{-\mathrm{y}}=-\mathrm{e}^{-\mathrm{x}}-\mathrm{e}^{\mathrm{x}}-\mathrm{x}^{2}+\mathrm{C}\)

4 \(\mathrm{e}^{\mathrm{y}}=\mathrm{e}^{-\mathrm{x}}+\mathrm{e}^{\mathrm{x}}+\mathrm{x}^{2}+\mathrm{C}\)

5 \(\mathrm{e}^{\mathrm{y}}=\mathrm{e}^{-\mathrm{x}}-\mathrm{e}^{\mathrm{x}}+\mathrm{C}\)

Differential Equation

87566 The curve, for which the area of the triangle formed by \(\mathrm{X}\)-axis, the tangent line at any point \(P\) and line OP is equal to \(a^{2}\), is given by

1 \(y=x-C x^{2}\)

2 \(x=C y \pm \frac{a^{2}}{y}\)

3 \(y=C x \pm \frac{a^{2}}{x}\)

4 None of these

Manipal UGET-2020]**#

Differential Equation

87567 The general solution of the differential equation
\(x \cos \frac{y}{x}(y d x+x d y)=y \sin \frac{y}{x}(x d y-y d x)\) is

1 \(\log (x y)=\log \cos \frac{x}{y}+C\)

2 \(\cos \left(\frac{y}{x}\right)=\frac{C}{x y}\)

3 \(\log (x y)=\log \sec \frac{x}{y}+C\)

4 \(x+y+C=0\)

Explanation:

(B) We have,
\(x \cos \left(\frac{y}{x}\right)(y d x+x d y)\)
\(=y \sin \frac{y}{x}(x d y-y d x)\)
\(\Rightarrow x y \sin \left(\frac{y}{x}\right) d y-y^{2} \sin \left(\frac{y}{x}\right) d x\)
\(=x y \cos \left(\frac{y}{x}\right) d x+x^{2} \cos \left(\frac{y}{x}\right) d y\)
\(\Rightarrow\left[x y \sin \left(\frac{y}{x}\right)-x^{2} \cos \left(\frac{y}{x}\right)\right] d y\)
\(=\left[x y \cos \left(\frac{y}{x}\right)+y^{2} \sin \left(\frac{y}{x}\right)\right] d x\)
\(\Rightarrow \frac{d y}{d x} =\frac{x y \cos \left(\frac{y}{x}\right)+y^{2} \sin \left(\frac{y}{x}\right)}{x y \sin \left(\frac{y}{x}\right)-x^{2} \cos \left(\frac{y}{x}\right)}\)
\(\Rightarrow \frac{d y}{d x} =\frac{\frac{y}{x} \cos \left(\frac{y}{x}\right)+\left(\frac{y}{x}\right)^{2} \sin \left(\frac{y}{x}\right)}{\frac{y}{x} \sin \left(\frac{y}{x}\right)-\cos \left(\frac{y}{x}\right)}\)
Put, \(y=x v \Rightarrow \frac{d y}{d x}=v+x \frac{d v}{d x}\)
\(\therefore \mathrm{v}+\mathrm{x} \frac{\mathrm{dv}}{\mathrm{dx}}=\frac{\mathrm{v} \cos \mathrm{v}+\mathrm{v}^{2} \sin \mathrm{v}}{\mathrm{v} \sin \mathrm{v}-\cos \mathrm{v}}\)
\(\Rightarrow \mathrm{x} \frac{\mathrm{dv}}{\mathrm{dx}}=\frac{2 \mathrm{v} \cos \mathrm{v}}{\mathrm{v} \sin \mathrm{v}-\cos \mathrm{v}} \Rightarrow \frac{\mathrm{v} \sin \mathrm{v}-\cos \mathrm{v}}{\mathrm{v} \cos \mathrm{v}} \mathrm{dv}=2 \frac{\mathrm{dx}}{\mathrm{x}}\)
\(\Rightarrow \int\left(\tan \mathrm{v}-\frac{1}{\mathrm{v}}\right) \mathrm{dv}=2 \int \frac{\mathrm{dx}}{\mathrm{x}}\)
\(\Rightarrow \log |\sec \mathrm{v}|-\log |\mathrm{v}|=2 \log |\mathrm{x}|+\log \left|\mathrm{C}_{1}\right|\)
\(\Rightarrow \log \left|\frac{\sec \mathrm{v}}{\mathrm{v}}\right|-\log |\mathrm{x}|^{2}=\log \left|\mathrm{C}_{1}\right|\)
\(\Rightarrow \log \left|\frac{\sec v}{v x^{2}}\right|=C_{1} \Rightarrow \frac{\sec \left(\frac{y}{x}\right)}{\frac{y}{x}\left(x^{2}\right)}=C_{1}\)
\(\Rightarrow \frac{\sec \left(\frac{y}{x}\right)}{x y}=C_{1} \Rightarrow \sec \left(\frac{y}{x}\right)=C_{1} x y\)
\(\Rightarrow \cos \left(\frac{\mathrm{y}}{\mathrm{x}}\right)=\frac{\mathrm{C}}{\mathrm{xy}}\)
\(\left(\because\right.\) where \(\left.\frac{1}{\mathrm{C}_{1}}=\mathrm{C}\right)\)

TS EAMCET-10.09.2020

Differential Equation

87568 Solution of the differential equation \(y^{\prime}=\) \(\frac{\left(x^{2}+y^{2}\right)}{x y}, y(1)=-2\) is given by

1 \(y_{2}^{2}=x^{2} \log x^{2}+4 x^{2}\)

2 \(y_{2}^{2}=x^{2} \log x-x^{2}\)

3 \(y^{2} x \log x^{2}+4 x^{2}\)

4 \(y^2=4 x^2 \log x^2+x^2\)

MHT CET-2021

Differential Equation

87569 The solution of the differential equation \(x \frac{d y}{d x}+y=\frac{1}{x^{2}}\) at \((1,2)\) is

1 \(x^{2} y+1=3 x\)

2 \(x^{2} y+1=0\)

3 \(x y+1=3 x\)

4 \(x^{2}(y+1)=3 x\)

5 \(x^{2} y=3 x+1\)

Kerala CEE-2011

Differential Equation

87570 The general solution of the differential equation \(\frac{d y}{d x}=e^{y}\left(e^{x}+e^{-x}+2 x\right)\) is

1 \(\mathrm{e}^{-\mathrm{y}}=\mathrm{e}^{\mathrm{x}}-\mathrm{e}^{-\mathrm{x}}+\mathrm{x}^{2}+\mathrm{C}\)

2 \(\mathrm{e}^{-\mathrm{y}}=\mathrm{e}^{-\mathrm{x}}-\mathrm{e}^{\mathrm{x}}-\mathrm{x}^{2}+\mathrm{C}\)

3 \(\mathrm{e}^{-\mathrm{y}}=-\mathrm{e}^{-\mathrm{x}}-\mathrm{e}^{\mathrm{x}}-\mathrm{x}^{2}+\mathrm{C}\)

4 \(\mathrm{e}^{\mathrm{y}}=\mathrm{e}^{-\mathrm{x}}+\mathrm{e}^{\mathrm{x}}+\mathrm{x}^{2}+\mathrm{C}\)

5 \(\mathrm{e}^{\mathrm{y}}=\mathrm{e}^{-\mathrm{x}}-\mathrm{e}^{\mathrm{x}}+\mathrm{C}\)

Differential Equation

87566 The curve, for which the area of the triangle formed by \(\mathrm{X}\)-axis, the tangent line at any point \(P\) and line OP is equal to \(a^{2}\), is given by

1 \(y=x-C x^{2}\)

2 \(x=C y \pm \frac{a^{2}}{y}\)

3 \(y=C x \pm \frac{a^{2}}{x}\)

4 None of these

Manipal UGET-2020]**#

Differential Equation

87567 The general solution of the differential equation
\(x \cos \frac{y}{x}(y d x+x d y)=y \sin \frac{y}{x}(x d y-y d x)\) is

1 \(\log (x y)=\log \cos \frac{x}{y}+C\)

2 \(\cos \left(\frac{y}{x}\right)=\frac{C}{x y}\)

3 \(\log (x y)=\log \sec \frac{x}{y}+C\)

4 \(x+y+C=0\)

Explanation:

(B) We have,
\(x \cos \left(\frac{y}{x}\right)(y d x+x d y)\)
\(=y \sin \frac{y}{x}(x d y-y d x)\)
\(\Rightarrow x y \sin \left(\frac{y}{x}\right) d y-y^{2} \sin \left(\frac{y}{x}\right) d x\)
\(=x y \cos \left(\frac{y}{x}\right) d x+x^{2} \cos \left(\frac{y}{x}\right) d y\)
\(\Rightarrow\left[x y \sin \left(\frac{y}{x}\right)-x^{2} \cos \left(\frac{y}{x}\right)\right] d y\)
\(=\left[x y \cos \left(\frac{y}{x}\right)+y^{2} \sin \left(\frac{y}{x}\right)\right] d x\)
\(\Rightarrow \frac{d y}{d x} =\frac{x y \cos \left(\frac{y}{x}\right)+y^{2} \sin \left(\frac{y}{x}\right)}{x y \sin \left(\frac{y}{x}\right)-x^{2} \cos \left(\frac{y}{x}\right)}\)
\(\Rightarrow \frac{d y}{d x} =\frac{\frac{y}{x} \cos \left(\frac{y}{x}\right)+\left(\frac{y}{x}\right)^{2} \sin \left(\frac{y}{x}\right)}{\frac{y}{x} \sin \left(\frac{y}{x}\right)-\cos \left(\frac{y}{x}\right)}\)
Put, \(y=x v \Rightarrow \frac{d y}{d x}=v+x \frac{d v}{d x}\)
\(\therefore \mathrm{v}+\mathrm{x} \frac{\mathrm{dv}}{\mathrm{dx}}=\frac{\mathrm{v} \cos \mathrm{v}+\mathrm{v}^{2} \sin \mathrm{v}}{\mathrm{v} \sin \mathrm{v}-\cos \mathrm{v}}\)
\(\Rightarrow \mathrm{x} \frac{\mathrm{dv}}{\mathrm{dx}}=\frac{2 \mathrm{v} \cos \mathrm{v}}{\mathrm{v} \sin \mathrm{v}-\cos \mathrm{v}} \Rightarrow \frac{\mathrm{v} \sin \mathrm{v}-\cos \mathrm{v}}{\mathrm{v} \cos \mathrm{v}} \mathrm{dv}=2 \frac{\mathrm{dx}}{\mathrm{x}}\)
\(\Rightarrow \int\left(\tan \mathrm{v}-\frac{1}{\mathrm{v}}\right) \mathrm{dv}=2 \int \frac{\mathrm{dx}}{\mathrm{x}}\)
\(\Rightarrow \log |\sec \mathrm{v}|-\log |\mathrm{v}|=2 \log |\mathrm{x}|+\log \left|\mathrm{C}_{1}\right|\)
\(\Rightarrow \log \left|\frac{\sec \mathrm{v}}{\mathrm{v}}\right|-\log |\mathrm{x}|^{2}=\log \left|\mathrm{C}_{1}\right|\)
\(\Rightarrow \log \left|\frac{\sec v}{v x^{2}}\right|=C_{1} \Rightarrow \frac{\sec \left(\frac{y}{x}\right)}{\frac{y}{x}\left(x^{2}\right)}=C_{1}\)
\(\Rightarrow \frac{\sec \left(\frac{y}{x}\right)}{x y}=C_{1} \Rightarrow \sec \left(\frac{y}{x}\right)=C_{1} x y\)
\(\Rightarrow \cos \left(\frac{\mathrm{y}}{\mathrm{x}}\right)=\frac{\mathrm{C}}{\mathrm{xy}}\)
\(\left(\because\right.\) where \(\left.\frac{1}{\mathrm{C}_{1}}=\mathrm{C}\right)\)

TS EAMCET-10.09.2020

Differential Equation

87568 Solution of the differential equation \(y^{\prime}=\) \(\frac{\left(x^{2}+y^{2}\right)}{x y}, y(1)=-2\) is given by

1 \(y_{2}^{2}=x^{2} \log x^{2}+4 x^{2}\)

2 \(y_{2}^{2}=x^{2} \log x-x^{2}\)

3 \(y^{2} x \log x^{2}+4 x^{2}\)

4 \(y^2=4 x^2 \log x^2+x^2\)

MHT CET-2021

Differential Equation

87569 The solution of the differential equation \(x \frac{d y}{d x}+y=\frac{1}{x^{2}}\) at \((1,2)\) is

1 \(x^{2} y+1=3 x\)

2 \(x^{2} y+1=0\)

3 \(x y+1=3 x\)

4 \(x^{2}(y+1)=3 x\)

5 \(x^{2} y=3 x+1\)

Kerala CEE-2011

Differential Equation

87570 The general solution of the differential equation \(\frac{d y}{d x}=e^{y}\left(e^{x}+e^{-x}+2 x\right)\) is

1 \(\mathrm{e}^{-\mathrm{y}}=\mathrm{e}^{\mathrm{x}}-\mathrm{e}^{-\mathrm{x}}+\mathrm{x}^{2}+\mathrm{C}\)

2 \(\mathrm{e}^{-\mathrm{y}}=\mathrm{e}^{-\mathrm{x}}-\mathrm{e}^{\mathrm{x}}-\mathrm{x}^{2}+\mathrm{C}\)

3 \(\mathrm{e}^{-\mathrm{y}}=-\mathrm{e}^{-\mathrm{x}}-\mathrm{e}^{\mathrm{x}}-\mathrm{x}^{2}+\mathrm{C}\)

4 \(\mathrm{e}^{\mathrm{y}}=\mathrm{e}^{-\mathrm{x}}+\mathrm{e}^{\mathrm{x}}+\mathrm{x}^{2}+\mathrm{C}\)

5 \(\mathrm{e}^{\mathrm{y}}=\mathrm{e}^{-\mathrm{x}}-\mathrm{e}^{\mathrm{x}}+\mathrm{C}\)

Differential Equation

87566 The curve, for which the area of the triangle formed by \(\mathrm{X}\)-axis, the tangent line at any point \(P\) and line OP is equal to \(a^{2}\), is given by

1 \(y=x-C x^{2}\)

2 \(x=C y \pm \frac{a^{2}}{y}\)

3 \(y=C x \pm \frac{a^{2}}{x}\)

4 None of these

Manipal UGET-2020]**#

Differential Equation

87567 The general solution of the differential equation
\(x \cos \frac{y}{x}(y d x+x d y)=y \sin \frac{y}{x}(x d y-y d x)\) is

1 \(\log (x y)=\log \cos \frac{x}{y}+C\)

2 \(\cos \left(\frac{y}{x}\right)=\frac{C}{x y}\)

3 \(\log (x y)=\log \sec \frac{x}{y}+C\)

4 \(x+y+C=0\)

Explanation:

(B) We have,
\(x \cos \left(\frac{y}{x}\right)(y d x+x d y)\)
\(=y \sin \frac{y}{x}(x d y-y d x)\)
\(\Rightarrow x y \sin \left(\frac{y}{x}\right) d y-y^{2} \sin \left(\frac{y}{x}\right) d x\)
\(=x y \cos \left(\frac{y}{x}\right) d x+x^{2} \cos \left(\frac{y}{x}\right) d y\)
\(\Rightarrow\left[x y \sin \left(\frac{y}{x}\right)-x^{2} \cos \left(\frac{y}{x}\right)\right] d y\)
\(=\left[x y \cos \left(\frac{y}{x}\right)+y^{2} \sin \left(\frac{y}{x}\right)\right] d x\)
\(\Rightarrow \frac{d y}{d x} =\frac{x y \cos \left(\frac{y}{x}\right)+y^{2} \sin \left(\frac{y}{x}\right)}{x y \sin \left(\frac{y}{x}\right)-x^{2} \cos \left(\frac{y}{x}\right)}\)
\(\Rightarrow \frac{d y}{d x} =\frac{\frac{y}{x} \cos \left(\frac{y}{x}\right)+\left(\frac{y}{x}\right)^{2} \sin \left(\frac{y}{x}\right)}{\frac{y}{x} \sin \left(\frac{y}{x}\right)-\cos \left(\frac{y}{x}\right)}\)
Put, \(y=x v \Rightarrow \frac{d y}{d x}=v+x \frac{d v}{d x}\)
\(\therefore \mathrm{v}+\mathrm{x} \frac{\mathrm{dv}}{\mathrm{dx}}=\frac{\mathrm{v} \cos \mathrm{v}+\mathrm{v}^{2} \sin \mathrm{v}}{\mathrm{v} \sin \mathrm{v}-\cos \mathrm{v}}\)
\(\Rightarrow \mathrm{x} \frac{\mathrm{dv}}{\mathrm{dx}}=\frac{2 \mathrm{v} \cos \mathrm{v}}{\mathrm{v} \sin \mathrm{v}-\cos \mathrm{v}} \Rightarrow \frac{\mathrm{v} \sin \mathrm{v}-\cos \mathrm{v}}{\mathrm{v} \cos \mathrm{v}} \mathrm{dv}=2 \frac{\mathrm{dx}}{\mathrm{x}}\)
\(\Rightarrow \int\left(\tan \mathrm{v}-\frac{1}{\mathrm{v}}\right) \mathrm{dv}=2 \int \frac{\mathrm{dx}}{\mathrm{x}}\)
\(\Rightarrow \log |\sec \mathrm{v}|-\log |\mathrm{v}|=2 \log |\mathrm{x}|+\log \left|\mathrm{C}_{1}\right|\)
\(\Rightarrow \log \left|\frac{\sec \mathrm{v}}{\mathrm{v}}\right|-\log |\mathrm{x}|^{2}=\log \left|\mathrm{C}_{1}\right|\)
\(\Rightarrow \log \left|\frac{\sec v}{v x^{2}}\right|=C_{1} \Rightarrow \frac{\sec \left(\frac{y}{x}\right)}{\frac{y}{x}\left(x^{2}\right)}=C_{1}\)
\(\Rightarrow \frac{\sec \left(\frac{y}{x}\right)}{x y}=C_{1} \Rightarrow \sec \left(\frac{y}{x}\right)=C_{1} x y\)
\(\Rightarrow \cos \left(\frac{\mathrm{y}}{\mathrm{x}}\right)=\frac{\mathrm{C}}{\mathrm{xy}}\)
\(\left(\because\right.\) where \(\left.\frac{1}{\mathrm{C}_{1}}=\mathrm{C}\right)\)

TS EAMCET-10.09.2020

Differential Equation

87568 Solution of the differential equation \(y^{\prime}=\) \(\frac{\left(x^{2}+y^{2}\right)}{x y}, y(1)=-2\) is given by

1 \(y_{2}^{2}=x^{2} \log x^{2}+4 x^{2}\)

2 \(y_{2}^{2}=x^{2} \log x-x^{2}\)

3 \(y^{2} x \log x^{2}+4 x^{2}\)

4 \(y^2=4 x^2 \log x^2+x^2\)

MHT CET-2021

Differential Equation

87569 The solution of the differential equation \(x \frac{d y}{d x}+y=\frac{1}{x^{2}}\) at \((1,2)\) is

1 \(x^{2} y+1=3 x\)

2 \(x^{2} y+1=0\)

3 \(x y+1=3 x\)

4 \(x^{2}(y+1)=3 x\)

5 \(x^{2} y=3 x+1\)

Kerala CEE-2011

Differential Equation

87570 The general solution of the differential equation \(\frac{d y}{d x}=e^{y}\left(e^{x}+e^{-x}+2 x\right)\) is

1 \(\mathrm{e}^{-\mathrm{y}}=\mathrm{e}^{\mathrm{x}}-\mathrm{e}^{-\mathrm{x}}+\mathrm{x}^{2}+\mathrm{C}\)

2 \(\mathrm{e}^{-\mathrm{y}}=\mathrm{e}^{-\mathrm{x}}-\mathrm{e}^{\mathrm{x}}-\mathrm{x}^{2}+\mathrm{C}\)

3 \(\mathrm{e}^{-\mathrm{y}}=-\mathrm{e}^{-\mathrm{x}}-\mathrm{e}^{\mathrm{x}}-\mathrm{x}^{2}+\mathrm{C}\)

4 \(\mathrm{e}^{\mathrm{y}}=\mathrm{e}^{-\mathrm{x}}+\mathrm{e}^{\mathrm{x}}+\mathrm{x}^{2}+\mathrm{C}\)

5 \(\mathrm{e}^{\mathrm{y}}=\mathrm{e}^{-\mathrm{x}}-\mathrm{e}^{\mathrm{x}}+\mathrm{C}\)

Differential Equation

87566 The curve, for which the area of the triangle formed by \(\mathrm{X}\)-axis, the tangent line at any point \(P\) and line OP is equal to \(a^{2}\), is given by

1 \(y=x-C x^{2}\)

2 \(x=C y \pm \frac{a^{2}}{y}\)

3 \(y=C x \pm \frac{a^{2}}{x}\)

4 None of these

Manipal UGET-2020]**#

Differential Equation

87567 The general solution of the differential equation
\(x \cos \frac{y}{x}(y d x+x d y)=y \sin \frac{y}{x}(x d y-y d x)\) is

1 \(\log (x y)=\log \cos \frac{x}{y}+C\)

2 \(\cos \left(\frac{y}{x}\right)=\frac{C}{x y}\)

3 \(\log (x y)=\log \sec \frac{x}{y}+C\)

4 \(x+y+C=0\)

Explanation:

(B) We have,
\(x \cos \left(\frac{y}{x}\right)(y d x+x d y)\)
\(=y \sin \frac{y}{x}(x d y-y d x)\)
\(\Rightarrow x y \sin \left(\frac{y}{x}\right) d y-y^{2} \sin \left(\frac{y}{x}\right) d x\)
\(=x y \cos \left(\frac{y}{x}\right) d x+x^{2} \cos \left(\frac{y}{x}\right) d y\)
\(\Rightarrow\left[x y \sin \left(\frac{y}{x}\right)-x^{2} \cos \left(\frac{y}{x}\right)\right] d y\)
\(=\left[x y \cos \left(\frac{y}{x}\right)+y^{2} \sin \left(\frac{y}{x}\right)\right] d x\)
\(\Rightarrow \frac{d y}{d x} =\frac{x y \cos \left(\frac{y}{x}\right)+y^{2} \sin \left(\frac{y}{x}\right)}{x y \sin \left(\frac{y}{x}\right)-x^{2} \cos \left(\frac{y}{x}\right)}\)
\(\Rightarrow \frac{d y}{d x} =\frac{\frac{y}{x} \cos \left(\frac{y}{x}\right)+\left(\frac{y}{x}\right)^{2} \sin \left(\frac{y}{x}\right)}{\frac{y}{x} \sin \left(\frac{y}{x}\right)-\cos \left(\frac{y}{x}\right)}\)
Put, \(y=x v \Rightarrow \frac{d y}{d x}=v+x \frac{d v}{d x}\)
\(\therefore \mathrm{v}+\mathrm{x} \frac{\mathrm{dv}}{\mathrm{dx}}=\frac{\mathrm{v} \cos \mathrm{v}+\mathrm{v}^{2} \sin \mathrm{v}}{\mathrm{v} \sin \mathrm{v}-\cos \mathrm{v}}\)
\(\Rightarrow \mathrm{x} \frac{\mathrm{dv}}{\mathrm{dx}}=\frac{2 \mathrm{v} \cos \mathrm{v}}{\mathrm{v} \sin \mathrm{v}-\cos \mathrm{v}} \Rightarrow \frac{\mathrm{v} \sin \mathrm{v}-\cos \mathrm{v}}{\mathrm{v} \cos \mathrm{v}} \mathrm{dv}=2 \frac{\mathrm{dx}}{\mathrm{x}}\)
\(\Rightarrow \int\left(\tan \mathrm{v}-\frac{1}{\mathrm{v}}\right) \mathrm{dv}=2 \int \frac{\mathrm{dx}}{\mathrm{x}}\)
\(\Rightarrow \log |\sec \mathrm{v}|-\log |\mathrm{v}|=2 \log |\mathrm{x}|+\log \left|\mathrm{C}_{1}\right|\)
\(\Rightarrow \log \left|\frac{\sec \mathrm{v}}{\mathrm{v}}\right|-\log |\mathrm{x}|^{2}=\log \left|\mathrm{C}_{1}\right|\)
\(\Rightarrow \log \left|\frac{\sec v}{v x^{2}}\right|=C_{1} \Rightarrow \frac{\sec \left(\frac{y}{x}\right)}{\frac{y}{x}\left(x^{2}\right)}=C_{1}\)
\(\Rightarrow \frac{\sec \left(\frac{y}{x}\right)}{x y}=C_{1} \Rightarrow \sec \left(\frac{y}{x}\right)=C_{1} x y\)
\(\Rightarrow \cos \left(\frac{\mathrm{y}}{\mathrm{x}}\right)=\frac{\mathrm{C}}{\mathrm{xy}}\)
\(\left(\because\right.\) where \(\left.\frac{1}{\mathrm{C}_{1}}=\mathrm{C}\right)\)

TS EAMCET-10.09.2020

Differential Equation

87568 Solution of the differential equation \(y^{\prime}=\) \(\frac{\left(x^{2}+y^{2}\right)}{x y}, y(1)=-2\) is given by

1 \(y_{2}^{2}=x^{2} \log x^{2}+4 x^{2}\)

2 \(y_{2}^{2}=x^{2} \log x-x^{2}\)

3 \(y^{2} x \log x^{2}+4 x^{2}\)

4 \(y^2=4 x^2 \log x^2+x^2\)

MHT CET-2021

Differential Equation

87569 The solution of the differential equation \(x \frac{d y}{d x}+y=\frac{1}{x^{2}}\) at \((1,2)\) is

1 \(x^{2} y+1=3 x\)

2 \(x^{2} y+1=0\)

3 \(x y+1=3 x\)

4 \(x^{2}(y+1)=3 x\)

5 \(x^{2} y=3 x+1\)

Kerala CEE-2011

Differential Equation

87570 The general solution of the differential equation \(\frac{d y}{d x}=e^{y}\left(e^{x}+e^{-x}+2 x\right)\) is

1 \(\mathrm{e}^{-\mathrm{y}}=\mathrm{e}^{\mathrm{x}}-\mathrm{e}^{-\mathrm{x}}+\mathrm{x}^{2}+\mathrm{C}\)

2 \(\mathrm{e}^{-\mathrm{y}}=\mathrm{e}^{-\mathrm{x}}-\mathrm{e}^{\mathrm{x}}-\mathrm{x}^{2}+\mathrm{C}\)

3 \(\mathrm{e}^{-\mathrm{y}}=-\mathrm{e}^{-\mathrm{x}}-\mathrm{e}^{\mathrm{x}}-\mathrm{x}^{2}+\mathrm{C}\)

4 \(\mathrm{e}^{\mathrm{y}}=\mathrm{e}^{-\mathrm{x}}+\mathrm{e}^{\mathrm{x}}+\mathrm{x}^{2}+\mathrm{C}\)

5 \(\mathrm{e}^{\mathrm{y}}=\mathrm{e}^{-\mathrm{x}}-\mathrm{e}^{\mathrm{x}}+\mathrm{C}\)