Solution of Linear Differential Equation
Differential Equation

87538 Let \(f:[0,1] \rightarrow R\) be such that \(f(x y)=f(x)\). \(f(y)\), for all \(x, y \in[0,1]\) and \(f(0) \neq 0\).If \(y=y\) ( \(x\) ) satisfies the differential equation, \(\frac{d y}{d x}=f(x)\) with \(y(0)=\) 1 , then \(y\left(\frac{1}{4}\right)+y\left(\frac{3}{4}\right)\) is equal to

1 5
2 3
3 2
4 4
Differential Equation

87540 The solution of differential equation \(y d x+\left(x-y^{2}\right) d y=0\)

1 \(\mathrm{e}^{\frac{y}{x}}=\sin \mathrm{x}+\mathrm{c}\)
2 \(y=c x \log x\)
3 \(x=\frac{y^{2}}{3}+\frac{c}{y}\)
4 \(\cos \left(\frac{y-2}{x}\right)=a\)
Differential Equation

87541 The solution of the differential equation \(\frac{d y}{d x}=\frac{x+y}{x}\) satisfying \(y(1)=1\) is

1 \(y=\log _{x-1} x+x\)
2 \(y=x \log x+x^{2}\)
3 \(y=x e^{x-1}\)
4 \(y=x \log x+x\)
Differential Equation

87542 If \(\frac{d y}{d x}=\frac{y+x \tan \frac{y}{x}}{x}\), then \(\sin \left(\frac{y}{x}\right)=\)

1 \(c x^{2}\)
2 \(\mathbf{c x}\)
3 \(\mathrm{cx}^{3}\)
4 \(\log \mathrm{x}\)
Differential Equation

87538 Let \(f:[0,1] \rightarrow R\) be such that \(f(x y)=f(x)\). \(f(y)\), for all \(x, y \in[0,1]\) and \(f(0) \neq 0\).If \(y=y\) ( \(x\) ) satisfies the differential equation, \(\frac{d y}{d x}=f(x)\) with \(y(0)=\) 1 , then \(y\left(\frac{1}{4}\right)+y\left(\frac{3}{4}\right)\) is equal to

1 5
2 3
3 2
4 4
Differential Equation

87540 The solution of differential equation \(y d x+\left(x-y^{2}\right) d y=0\)

1 \(\mathrm{e}^{\frac{y}{x}}=\sin \mathrm{x}+\mathrm{c}\)
2 \(y=c x \log x\)
3 \(x=\frac{y^{2}}{3}+\frac{c}{y}\)
4 \(\cos \left(\frac{y-2}{x}\right)=a\)
Differential Equation

87541 The solution of the differential equation \(\frac{d y}{d x}=\frac{x+y}{x}\) satisfying \(y(1)=1\) is

1 \(y=\log _{x-1} x+x\)
2 \(y=x \log x+x^{2}\)
3 \(y=x e^{x-1}\)
4 \(y=x \log x+x\)
Differential Equation

87542 If \(\frac{d y}{d x}=\frac{y+x \tan \frac{y}{x}}{x}\), then \(\sin \left(\frac{y}{x}\right)=\)

1 \(c x^{2}\)
2 \(\mathbf{c x}\)
3 \(\mathrm{cx}^{3}\)
4 \(\log \mathrm{x}\)
Differential Equation

87538 Let \(f:[0,1] \rightarrow R\) be such that \(f(x y)=f(x)\). \(f(y)\), for all \(x, y \in[0,1]\) and \(f(0) \neq 0\).If \(y=y\) ( \(x\) ) satisfies the differential equation, \(\frac{d y}{d x}=f(x)\) with \(y(0)=\) 1 , then \(y\left(\frac{1}{4}\right)+y\left(\frac{3}{4}\right)\) is equal to

1 5
2 3
3 2
4 4
Differential Equation

87540 The solution of differential equation \(y d x+\left(x-y^{2}\right) d y=0\)

1 \(\mathrm{e}^{\frac{y}{x}}=\sin \mathrm{x}+\mathrm{c}\)
2 \(y=c x \log x\)
3 \(x=\frac{y^{2}}{3}+\frac{c}{y}\)
4 \(\cos \left(\frac{y-2}{x}\right)=a\)
Differential Equation

87541 The solution of the differential equation \(\frac{d y}{d x}=\frac{x+y}{x}\) satisfying \(y(1)=1\) is

1 \(y=\log _{x-1} x+x\)
2 \(y=x \log x+x^{2}\)
3 \(y=x e^{x-1}\)
4 \(y=x \log x+x\)
Differential Equation

87542 If \(\frac{d y}{d x}=\frac{y+x \tan \frac{y}{x}}{x}\), then \(\sin \left(\frac{y}{x}\right)=\)

1 \(c x^{2}\)
2 \(\mathbf{c x}\)
3 \(\mathrm{cx}^{3}\)
4 \(\log \mathrm{x}\)
Differential Equation

87538 Let \(f:[0,1] \rightarrow R\) be such that \(f(x y)=f(x)\). \(f(y)\), for all \(x, y \in[0,1]\) and \(f(0) \neq 0\).If \(y=y\) ( \(x\) ) satisfies the differential equation, \(\frac{d y}{d x}=f(x)\) with \(y(0)=\) 1 , then \(y\left(\frac{1}{4}\right)+y\left(\frac{3}{4}\right)\) is equal to

1 5
2 3
3 2
4 4
Differential Equation

87540 The solution of differential equation \(y d x+\left(x-y^{2}\right) d y=0\)

1 \(\mathrm{e}^{\frac{y}{x}}=\sin \mathrm{x}+\mathrm{c}\)
2 \(y=c x \log x\)
3 \(x=\frac{y^{2}}{3}+\frac{c}{y}\)
4 \(\cos \left(\frac{y-2}{x}\right)=a\)
Differential Equation

87541 The solution of the differential equation \(\frac{d y}{d x}=\frac{x+y}{x}\) satisfying \(y(1)=1\) is

1 \(y=\log _{x-1} x+x\)
2 \(y=x \log x+x^{2}\)
3 \(y=x e^{x-1}\)
4 \(y=x \log x+x\)
Differential Equation

87542 If \(\frac{d y}{d x}=\frac{y+x \tan \frac{y}{x}}{x}\), then \(\sin \left(\frac{y}{x}\right)=\)

1 \(c x^{2}\)
2 \(\mathbf{c x}\)
3 \(\mathrm{cx}^{3}\)
4 \(\log \mathrm{x}\)