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question_answer1) If m and n are order and degree of the equation\[{{\left( \frac{{{d}^{2}}y}{d{{x}^{2}}} \right)}^{5}}+4.\frac{{{\left( \frac{{{d}^{2}}y}{d{{x}^{2}}} \right)}^{3}}}{\frac{{{d}^{3}}y}{d{{x}^{3}}}}+\frac{{{d}^{3}}y}{d{{x}^{3}}}={{x}^{2}}-1\], then find value of\[m\text{ }+\text{ }n\].
question_answer2) If the slope of the curve \[y=\frac{ax}{b-x}\] at the point (1, 1) is 2, then find the value of\[a+b\].
question_answer3) Find the degree of differential equation\[{{\left[ 1+2{{\left( y' \right)}^{2}} \right]}^{3/2}}=5y\].
question_answer4) If \[x\,\,dy\,\,=y\,dx+{{y}^{2}}\,dy\] and \[y\left( 1 \right)=1\] then find the value of\[y\left( -3 \right)\].
question_answer5) If \[\phi \,\left( x \right)=\phi '\left( x \right),\phi \left( 1 \right)=2\] and \[\phi \left( 3 \right)=k{{e}^{2}}\] then find k.
question_answer6) Equation of the curve passing through the point (1, 2) such that the intercept on the x-axis cut off between the tangent and origin is twice the abscissa is given by \[xy\text{ }=\text{ }k\] then find k.
question_answer7) Find the order of the differential equation, whose general solution \[y={{c}_{1}}{{e}^{x}}+{{c}_{2}}{{e}^{2x}}+{{c}_{3}}{{e}^{3x}}+{{c}_{4}}{{e}^{x+{{c}_{5}}}},\] (Where are \[{{c}_{1}},{{c}_{2}},{{c}_{3}},{{c}_{4}},{{c}_{5}}\] are arbitrary constants)
question_answer8) Find the sum of order and degree of differential equation of all tangent lines to the parabola \[{{x}^{2}}=4y\].
question_answer9) Equation of curve through (2, 2) satisfying \[\left( 1-{{x}^{2}} \right)\frac{dy}{dx}+xy=5x\] is \[{{\left( y-5 \right)}^{2}}+k\left( 1-{{x}^{2}} \right)=0\] then find k.
question_answer10) The slope at any point of a curve \[y=f\left( x \right)\] is given by \[\frac{dy}{dx}=3{{x}^{2}}\] and it passes through\[\left( -1,1 \right)\]. The equation of the curve is \[y={{x}^{n}}+2\] then find n.
question_answer11) The solution of \[\frac{dy}{dx}+\frac{y}{\left( 1+{{x}^{2}} \right)}=\frac{{{e}^{{{\tan }^{-1}}}}}{1={{x}^{2}}}\] is \[{{e}^{k\,{{\tan }^{-1}}x}}+2c\] then find k.
question_answer12) The solution of the differential equation \[\left( 2x-10{{y}^{3}} \right)\frac{dy}{dx}+y=0\] is \[x{{y}^{n}}=2{{y}^{m}}+C\] then find \[n\text{ }+\text{ }m.\]
question_answer13) The solution of \[\frac{dy}{dx}=\frac{6{{x}^{2}}}{\left( 2y+\cos y \right)},y\left( 1 \right)=0,\] is \[{{y}^{2}}+\sin y=k{{x}^{3}}-2\] then find k.
question_answer14) If the integral factor of equation \[\left( {{x}^{2}}+1 \right)\frac{dy}{dx}+2xy={{x}^{2}}-1,\] is \[{{x}^{2}}\,+k\] then find k.
question_answer15) The equation of the curve through the point (3, 2) and whose slope is \[\frac{{{x}^{2}}}{y+1},\]is \[\frac{{{y}^{2}}}{2}+y=\frac{{{x}^{3}}}{3}-k\] then find k.
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